Algebra II Foundations

Transcription

1 AIIF Algebra II Foundations Non Linear Functions Teacher Manual

2 Table of Contents Lesson Page Lesson 1: Introduction to Quadratic Functions...1 Lesson : The Quadratic Formula...17 Lesson 3: Graphing Quadratic Functions and Their Applications...35 Lesson 4: Power Functions...63 Lesson 5: Inverse Variation...89 Lesson 6: Exponential Functions Lesson 7: Step Functions Lesson 8: Miscellaneous Non Linear Functions Assessments CREDITS Author: Contributors: Graphic Design: Dennis Goyette and Danny Jones Robert Balfanz, Dorothy Barry, Leonard Bequiraj, Stan Bogart, Robert Bosco, Carlos Burke, Lorenzo Hayward, Vicki Hill, Winnie Horan, Donald Johnson, Kay Johnson, Karen Kelleher, Kwan Lange, Dennis Leahy, Song-Yi Lee, Hsin-Jung Lin, Guy Lucas, Ira Lunsk, Sandra McLean, Hemant Mishra, Glenn Moore, Linda Muskauski, Tracy Morrison, Jennifer Prescott, Gerald Porter, Steve Rigefsky, Ken Rucker, Stephanie Sawyer, Dawne Spangler, Fred Vincent, Maria Waltemeyer, Teddy Wieland Gregg M. Howell Copyright 009, The Johns Hopkins University, on behalf of the Center for Social Organization of Schools. All Rights Reserved. CENTER FOR SOCIAL ORGANIZATION OF SCHOOLS Johns Hopkins University 3003 N. Charles Street Suite 00 Baltimore, MD fax All rights reserved. Student assessments, Cutout objects, and transparencies may be duplicated for classroom use only; the number is not to exceed the number of students in each class. No other part of this document may be reproduced, in any form or by any means, without permission in writing from the publisher. Transition to Advanced Mathematics contains Internet website IP (Internet Protocol) addresses. At the time this manual was printed, the website addresses were checked for both validity and content as it relates to the manual s corresponding topic. The Johns Hopkins University, and its licensors is not responsible for any changes in content, IP addresses changes, pop advertisements, or redirects. It is further recommended that teachers confirm the validity of the listed addresses if they intend to share any address with students.

3 Non Linear Functions Planning Document AIIF Page i Planning Document: Non-Linear Functions Overview The types of non-linear functions include: Quadratic functions Power functions Inverse variation functions Exponential functions Step functions Absolute value functions Circles (domain restricted to be a function) Piece-wise functions The number of total suggested days for the unit is 18. Adjustments may be needed based on student performance during the unit and amount of time available until the end of the semester. Vocabulary Square/squaring Quadratic equation Square root Minimum point Maximum point Standard form of a quadratic function General form of a quadratic function Parabola Solutions Quadratic formula Discriminant Double root Break-even point Symmetry Vertical line symmetry Vertex x-coordinate y-coordinate Quadratic regression Multiples Power function Even function Odd function Direct variation Inverse variation Constant of proportionality Base Exponent Exponential function Growth Decay Exponential regression Rise Run int() function Greatest integer Floor function Smallest integer Ceiling function Binary number system Absolute value function Dilation Vertical line test General equation of a circle Piece-wise function Material List Student journal Setting the Stage transparencies Dry-erase boards Markers and erasers Chart paper Graphing calculators Calculator view screen Blank transparencies Lesson specific transparencies Overhead projector Construction paper Poster paper Colored pencils

4 AIIF Page ii The following table contains lesson name, timeline, summary of concepts covered, and the Essential Question(s) for each lesson. Lesson Timeline Concepts Covered Essential Question(s) Introduction to Quadratic Functions The Quadratic Formula Graphing Quadratic Functions and Their Applications 1 Day Write and solve simple quadratic equations Use the graphing calculator to find vertex, x- intercepts, and to draw a graph Days Standard form of a quadratic function Quadratic formula Discriminant Solve quadratic functions using the quadratic formula y-intercept Applications of quadratic functions 3 Days Line of symmetry Vertex Graphing Applications of quadratic functions Quadratic regression Power Functions Days Power function format Odd and even functions Graphing power functions Transformations involving power functions Applications of power functions Inverse Variation Days Write equations involving direct variation Constant of proportionality Write equations involving inverse (indirect) variation Graph direct and inverse variation Identify inverse variation phrases Exponential Functions 3 Days Identify exponential functions Growth and decay Exponential applications Graph exponential functions How does the process of squaring relate to quadratic functions? How can the quadratic formula be used to solve real-world applications? How can quadratic functions and applications of quadratic functions be graphed? Do power functions have patterns that can be used when solving and graphing them? How does inverse variation affect real world application problems? How does direction variation affect real world application problems? How do exponential functions behave in real-world applications?

5 Non Linear Functions Planning Document Step Functions Days Rise and run of a step function Floor step function Ceiling step function Graph step functions using the graphing calculator and the int() function Applications involving step functions Miscellaneous Non- Linear Functions 3 Days Absolute value functions o Transformations involving absolute value functions o Graph absolute value functions Circles o Equations for circles o Restrict domain of a circle equation o Solve circle equations for y o Graph circle equations using center, radius, and intercepts Piece-wise functions o Write piece-wise functions o Graph piece-wise functions o Applications of piece-wise functions AIIF Page iii How do step functions apply to real-world applications? How do absolute value functions, piece wise functions, and circle equations apply to real world applications?

6 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 1 Lesson 1: Introduction to Quadratic Functions Objectives Students will be able to identify quadratic functions Students will be able to use the squaring process to create quadratic functions Students will be able to solve simple quadratic functions by taking the square root Students will be able to graph quadratic functions Students will be able to determine the maximum or minimum of a quadratic function Essential Questions How does the process of squaring relate to quadratic functions? Tools Student Journal Setting the Stage transparency Dry erase boards, markers, erasers Chart paper Graphing calculators Warm Up Problems of the Day Number of Days 1 Day Vocabulary Quadratic equation Maximum point Horizontal translation Square/Squaring Minimum point Square root Vertical translation Notes Prior to teaching, you will need to prepare transparencies from the master hard copies supplied in this manual. At the end of each lesson in Algebra II Foundations there are Practice Exercises, Outcome Sentences, and a small quiz. The authors suggest that teachers use these tools as needed and as time allows.

7 AIIF Page Teacher Reference Setting the Stage Place the Setting the Stage transparency on the overhead projector but cover it at first. Give each student a piece of string 10 inches long (the grid squares on the dry erase boards are 5/16 of an inch square). Eight squares make.5 inches. Have the students make a square from their piece of string on their dry erase board (grid side) that is 8 units by 8 units. After the class has made a square, 8 X 8, uncover the transparency. Ask the class the following questions or something similar: What are the lengths of the sides of your square? What is the area of your square? What is the formula for the area of a square? Write an equation that represents the area of your square (we are looking so see if the students come up with something like x = 64 or s = 64. Some students might write A = 64 square units.) The key is that the students see that we need to square the length of a side of the square to get the area. What is the exponent of your variable in your equation? What is the term we use when we raise something to the second power? (We are looking for students to use the term square or squaring.) What is the opposite operation called? (We are looking for students to use the term square root.) Tell the class that equations/functions that have the independent variable to the second power, or squared, are called quadratic equations or quadratic functions respectively. You might want to tell the class that the term quadratic comes from the Latin word quadratus, which means square. Ask the class how the Latin word quadratus is related to the term quadratic. The key concept is that quadratic has a variable term with the highest exponent being, meaning squared. Some Latin words you may want to use are quadratum for square, or quad meaning 4. You could ask the class to name mathematic terms that start with the word quad like quadrilateral, quadrants, quadrangle, and quadruple.

8 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 3 Setting the Stage Transparency

9 AIIF Page 4 Teacher Reference Activity 1 In this activity, students will create, work with, and solve simple quadratic equations. Think aloud as you model the following problem: The area of a square yard is 49 square meters. What are the dimensions of the yard? What is the perimeter of the yard? Even though some students might be able to give the answer right away, make sure to model how to properly set up and solve these types of equations. You could refer to the table method in the Solving Equations unit. Some of the concepts you should model are: Label variable(s) (here we would label or state that the length equals the width which we could label as s for the length of the sides of the square yard.) Write an equation, using your variable(s) representing the problem (here, using our variable we would have s = 49.) Solve the equation, make sure the answer contains appropriate units (the solution is s = 7 meters.) Have a volunteer model the following problem, on the overhead projector, while the class follows along on their dry-erase boards: A number squared is 36. If the students come up with only one solution, ask, "Is there any other number that when squared equals 36?" The key is for the students to understand that both a negative six and a positive six, when squared, would equal 36. Now ask the class, "Why was there only one answer for the unknown in the problem related to the dimensions of the yard with an area of 49 square meters?" The goal is for students to differentiate between problems with just numbers and variables and real world applications. Sometimes answers just don t make sense in the real world even though mathematically they are correct. For example, you cannot measure using negative numbers. Have another volunteer model the following problem, on the overhead projector, while the class follows along, using dry erase boards: The square of the sum of a number and three is sixteen. This problem is a bit more complicated than the previous one. For the class to write the correct equation you may want to ask guiding questions such as What is being squared?, "What letter should be used for the variable", and What must be calculated first, the sum or the square? These questions are to help the students come up with the equation( x + 3) = 16. The questions are also to determine how the students will handle solving a quadratic equation. You will want to let the students explore different strategies to determine how to solve this equation and then talk about their strategies. The students can refer back to the table method. Have the students work individually on exercises 1 through 6. After the students have completed the exercises have them pair up with another student to go through the exercises together and verify answers. If they disagree on any of the answers, have them check with another student pair or pairs until they agree. Have volunteers share their answers with the class on any exercises not agreed upon. Circulate while the students are working to ask guiding questions and encourage the students. Note: exercises 5 and 6 may need some additional explanation.

10 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 5 Activity 1 SJ Page 1 For the following exercises, write a matching quadratic equation then solve the equation and answer each question. 1. The square of a number is sixty-four. What are the numbers that make the equation true? The quadratic equation is n = 64. Solving this equation we get that the number can be 8 or 8.. The square of the difference of a number and six is one hundred twenty-one. What are the numbers? The quadratic equation is( n 6) = 11. The answers are n = 5 and n = The area of a square piece of paper is 144 square inches. What are the lengths of the sides of the square piece of paper? The quadratic equation is n = 144. The lengths of the sides of the square are 1 inches. Crop circles are patterns created by the flattening of 4. The area of a crop circle is approximately 157 square feet. crops such as wheat, barley, Approximately what is the size of the diameter of the circle? Use rapeseed, rye, corn, linseed, and soy into circles. The term 3.14 forπ. The quadratic equation isπ r = 157. Solving this was first used by researcher equation we get r 0. Because the radius is approximately 0 Colin Andrews to describe feet, this makes the diameter of the crop circle about 40 feet. simple circles he was 5. A company s cost can be determined by the sum of the variable costs and the fixed costs. Namely, C(x) = variable costs + fixed researching. costs. It has been determined that the company s variable costs are the cost of producing one unit times the square of the number of units. a. If it costs $11.00 to produce a single unit of the product and the company's fixed costs are $1,000.00, write an equation, in function notation, that represents the total costs of producing x units of the product. The quadratic cost function isc(x) = 11x b. If total costs are $,751,000.00, how many units of the product are produced? There are 500 units of the product produced. 6. A company makes a product. The company has determined the approximate cost to produce a single unit of the product. The company has fixed costs of $500. The company also knows that it costs $50,500 to produce 100 units of the product. The engineering department s research shows that the variable portion of the cost function is the cost to produce a single unit times the square of the number of units produced. That is, variable costs = cx. Write a cost function, C(x), which represents the total cost of producing x units of the product. Use the information given to determine the cost to produce one unit of the product. The quadratic equation isc(x) = cx The cost per unit of the product is $ = c = 10000c = 10000c = 5 = c 10000

11 AIIF Page 6 Teacher Reference Activity In this activity, students will investigate the graphs of quadratic functions using the classroom graphing calculator. Use the Parallel Modeling strategy to model graphing y = x while students model the same or a similar equation. For this first equation, have the students model the same equation with you. You may want to prepare a list of questions to ask the students after the graph is displayed on the graphing calculator view screen and on the students' graphing calculators. Here are some questions that you could ask: What is the shape of the graph? What do you notice about the portion of the graph to the left of the y-axis compared to the portion to the right of the y axis? What do you know about minimum and maximum values? Does the graph have a minimum or maximum value? Have a volunteer record the responses and characteristics on the board while the students record the responses and characteristics in their student journals. Next, have a volunteer model graphing the equation y = x in the front of the class. Have a second volunteer, along with the other students, list the similarities and differences between this equation and y = x. The goal is that the students notice that the graphs are exactly the same except that y = x is flipped, or reflected, about the x axis and is the opposite of y = x. Now have another volunteer model the equation y = x while the class models y = 3x. Have a second volunteer record the class' responses about the similarities between the very first equation graphed, y = x, and the equation they have graphed now. You may want students to display both graphs at the same time to make the comparison easier. Have the class get back in their pairs from Activity 1. Give each pair a piece of chart paper. Have the pairs complete a similar comparison for y = x and y = 3x. You may decide to have the students display all 3 graphs on their calculators at the same time (y = x, y = 3x, y = 3x, and maybe even y = x ). Have the pairs list the similarities and differences between the graphs on their chart paper and draw a copy of the graphs from the graphing calculator to the chart paper. Give the students five to seven minutes to complete the task and then have the class display their chart paper along the walls of the classroom. The class can then walk around checking other pair's graphs and list of similarities and differences. Lead a class discussion on the similarities and differences that the class listed. You may want to show the class how to use the graphing calculator capabilities to find the maximum or minimum y value of each graph. The graphing calculator can display the coordinates of the minimum or maximum value. The steps and screen shots below pertain specifically to the TI-83 or 84 Plus graphing calculator. Steps to Find the Maximum/Minimum Press the ND key followed by the TRACE key to display the CALCULATE menu (assuming the equation has been entered into the Y= editor and the graph is displayed). Press 3 for minimum or 4 for maximum Use the arrow keys to move the blinking cursor to the left of the minimum or maximum point on the graph (calculator is looking for the Left Bound) and press the ENTER key. Use the arrow keys to move the blinking cursor to the right of the minimum or maximum point on the graph (calculator is looking for the Right Bound) and press the ENTER key.

12 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 7 Use the arrow keys to move the blinking cursor to the minimum or maximum point on the graph (calculator is looking for your Guess?) and press the ENTER key. Minimum or maximum coordinates are displayed at the bottom of the screen The screen shots below are for the above steps. Mention the TRACE key to the students which will allow them to trace along the graphs. Also mention that the up and down arrow keys will allow students to switch between graphs in order to distinguish which graph represents which equation. Model, or have a student model, graphing the equation y = x + 1 while the class parallels with y = x +. Ask the students, "What have you noticed about the minimum value of y and its coordinates?" The students should realize that the minimum value of y and its coordinates have been translated vertically by two units or one unit, respectively. You could also model an equation where the vertical translation is negative so that the students understand that the constant added will tell them what the minimum or maximum value is as well as its coordinates. Now model, or have a student model, graphing the equation y = (x ) while the class parallels with y = (x 1). Ask the students, "What have you noticed about the minimum value of y and its coordinates?" The students should realize that the minimum value of y and its coordinates have been translated horizontally by two units or one unit, respectively. You could also model an equation where the horizontal translation is negative so that the students understand that the constant added with x before it is squared will tell them the x coordinate of the minimum value. Now model, or have a student model, graphing the equation y = (x ) + while the class parallels with y = (x 1) + 1. Ask the students, "What have you noticed about the minimum value of y and its coordinates?" The students should realize that the minimum value of y and its coordinates have been translated both vertically and horizontally. You could also model equations where the horizontal and vertical translations are a combination of positive and negative values.

13 AIIF Page 8 As a last modeling example, display the graph of the equation y = (x +1) 4. Model to the class how the coordinates of the vertex can be used to determine the equation for the graph. From the previous examples, we know that adding/subtracting a constant from x before squaring creates a horizontal translation while adding/subtracting a constant from the squared term creates a vertical translation. Show the students how using the coordinates of the vertex from the graph, they can obtain the equation y = (x +1) 4. The graph is displayed below. Have the students work in pairs on Exercises 1 through 10. After the students have completed the exercises, have each pair join another pair and compare their findings. Have the students include in their work contributions given by the other student pair in their group. Bring the class together and have volunteers share their results with the class.

14 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 9 Activity SJ Page In this activity, you will investigate the graphs of various quadratic equations. For the following exercises, find the coordinates of the minimum or maximum value and state the minimum or maximum y value, all x intercepts and y intercepts, and make a sketch of the graph in the grid provided. The exercises are set up in most cases to draw two graphs per grid. You may also want to display the two graphs simultaneously on your graphing calculator as well. NOTE: Displaying table values on your graphing calculator may help you to draw the graph y = x Minimum y value is 0 and the coordinates of the minimum value is (0, 0). Only x intercept and y intercept is (0, 0).. y = 4x Minimum y value is 0 and the coordinates of the minimum value is (0, 0). Only x intercept and y intercept is (0, 0). 3. y = x + 3 Maximum y value is 3 and the coordinates of the maximum value is (0, 3); y intercept is (0, 3). There are two x intercepts: ( 3,0) and ( 3,0), or approximately (1.73,0) and ( 1.73, 0). The students may be familiar with the decimals rather than the radicals. 4. y = x Maximum y value is and the coordinates of the maximum value is (0, ); y-intercept is (0, ). There are no x intercepts.

15 AIIF Page 10 SJ Page 3 y = x ( ) Minimum y value is 0 and the coordinates of the minimum value is (0,0); y intercept is (0, 9). Only x intercept is ( 3, 0). y = x 6. ( ) Minimum y value is 0 and the coordinates of the minimum value is (, 0); y intercept is (0, 4). Only x intercepts is (, 0). y = x ( ) Minimum y value is 3 and the coordinates of the minimum value is (, 3); y intercept is (0, 7). There are no x intercepts. 8. y ( x ) = 3 Minimum y value is 3 and the coordinates of the minimum value is (, 3); y intercept is (0, 1). There are two x intercepts: approximately (3.73, 0) and (0.7, 0).

16 Non Linear Functions Lesson 1: Introduction to Quadratic Functions 9. For the given graph, identify the coordinates of the minimum point, x intercepts, y intercept, and equation representing the graph. AIIF Page 11 SJ Page 4 The minimum point has the coordinates (, 9); the equation of the graph is y = (x +) 9; the x intercepts are ( 5, 0) and (1, 0); the y intercept is (0, 5). 10. From Exercises 1 through 9, what conclusions and characteristics can you make about the graphs of quadratic equations? Answers will vary. Sample response might be: The larger the number in front of x, the more narrow the graph; positive numbers in the front of x give a minimum value while negative numbers in the front of x give a maximum value; adding or subtracting a number from the square term vertically translates the graph that number of units up or down; adding and subtracting a number from the variable and then squaring it translates the graph horizontally by that number of units to the left or right.

17 AIIF Page 1 SJ Page 5 Practice Exercises Solve each of the following. 1. The square of a number is one hundred forty-four. Write a quadratic equation and then solve for the unknown number(s). The equation is n = 144; the numbers are 1 and 1.. The square of the sum of a number and nine is one hundred sixty-nine. Write a quadratic equation and then solve for the unknown number(s). The equation is ( n+ 9) = 169; the numbers are 4 and. 3. The Sparkling Diamonds jewelry store sold a diamond studded bracelet and made a profit of $196. The profit is based on the cost of the necklace to the store. How much did the necklace cost the store if profit C C is determined by the equation P= C =, where P is the profit and C is the cost of the item? The necklace cost the store $ Graph the quadratic equation y = 3x + 1 on the grid supplied below. Label all intercepts and determine the maximum or minimum point. The y intercept is (0, 1); the x intercepts are (, 0) and (, 0). The maximum point is (0, 1). 5. Graph the quadratic equation ( ) y = x 4 on the grid supplied below. Label all intercepts and determine the maximum or minimum point. The y intercept is (0, 16); the x intercept is (4, 0). The minimum point is (4, 0)

18 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 13 SJ Page 6 y = x Label all intercepts and determine the maximum or minimum point. The y intercept is (0, 33); the x intercepts are (3, 0) and (11, 0). The minimum point is (7, 16). 6. Graph the quadratic equation ( ) 7. For the given graph, identify the minimum point, x intercepts, y intercept, and equation representing the graph. The minimum point is ( 3, 4), the x intercepts are ( 1, 0) and ( 5, 0), the y intercept is (0, 5), the equation of the graph is y = (x + 3) How does the process of squaring relate to quadratic functions? Answers will vary. Sample response: Because the term quadratic comes from a Latin word meaning "to square," squaring the x variable makes the function quadratic compared to something else like linear.

19 AIIF Page 14 SJ Page 7 Outcome Sentences To solve a quadratic equation I know a quadratic equation will have a minimum when I know a quadratic equation will have a maximum when The minimum or maximum of a quadratic equation can be determined by I can use the graphing calculator to I would like to find out more about I now understand I still have a question about

20 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 15 Teacher Reference Lesson 1 Quiz Answers 1. y-intercepts (0, 3); the x-intercepts are (4, 0) and ( 8, 0). The minimum point is (, 36).. The equation is ( n 6) = 100. The unknown numbers are 16 and 4. 3a. 5 inches 3b. 6 inches

21 AIIF Page 16 Lesson 1 Quiz Name: 1. Graph the quadratic equation ( ) minimum point. y = x+ 36. Label all intercepts and determine the maximum or. The square of the difference of a number and six is one hundred. Write a quadratic equation and then solve for the unknown number(s). 3. Oatmeal is a great nutritious breakfast on a cold morning. Most oatmeal containers are cylindrical. The volume of a cylindrical container is given byv = π r h, where r is the radius of the container, and h is the height of the container. Find the radius of an oatmeal container when: a. The volume is cubic inches and the height is 9.5 inches. Approximate π with Round answer to the nearest quarter inch. b. The volume is 678 cubic inches and the height is 6 inches. Assume 3.14 for pi. Round answer to the nearest quarter inch.

22 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 17 Lesson : The Quadratic Formula Objectives Students will be able to use technology to understand, simplify, and solve quadratic equations. Students will be able to write quadratic equations in general form in order to use the quadratic formula. Students will be able to use the discriminant to determine the nature of the roots for a quadratic equation. Students will be able to use the quadratic formula to solve real world applications. Essential Questions How can the quadratic formula be used to solve real-world applications? Tools Student Journal Setting the Stage transparency Blank transparencies Overhead projector Dry erase boards, markers, erasers Graphing calculators Warm Up Problems of the Day Number of Days days Vocabulary Standard form of a quadratic function General form of a quadratic functioin Quadratic formula Zeros Parabola Discriminant Break-even point Solution Double root x-intercept

23 AIIF Page 18 Teacher Reference Setting the Stage Talk with students about how computer programmers and designers are often trying to determine a quicker method for computers to work, so that as you use programs or play video games the lag time is as short as possible. This is done in basically two ways. One way is to make the hardware in the computer, such as the computer chips and circuits, faster by physically making them smaller and with better conducting material so that the electronic information can travel quickly within the computer. Another method is to design the software (program) with the least amount of program language instructions as possible. Mathematics has a major influence in both of these areas. The geometry, physics, and chemistry behind making the hardware smaller is all built on math formulas. Often in the computer program, functions and equations are entered so that the computer calculates and outputs a display as a number or colored pixel on the screen. Display the Setting the Stage transparency 1. For example, a particular software program may need to display a line or lines across the screen. In order to do this the program would need to calculate the distance between two points. Which method do you think would save time and allow the program to run faster? Explain. Display the second transparency when necessary to visually illustrate the two points and the line, representing the distance, between the two points. Talk with students that in math we often find methods to make steps shorter. In many cases you have learned how to shorten the steps yourself. Today, you are going to encounter a short method to solve a quadratic equation and skip many steps that would take weeks to teach you. By recognizing what values to input, you will be able to calculate the output that solves the equation without using any solving techniques like you do for linear equation. Enjoy the short cut!!!

24 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 19 Setting the Stage Transparency 1 Method 1 Step 1: Input x1, x, y1, and y Step : Calculate y y1 Step 3: Square the value in Step Step 4: Calculate x x1 Step 5: Square the value in Step 4 Step 6: Sum the values from Step 3 and Step 5 Step 7: Determine the square root of the value in Step 6 Step 8: Output the value from Step 7 Method Step 1: Input x1, x, y1, and y Step : Calculate ( ) ( ) x x + y y 1 1 Step 3: Output the value from Step

25 AIIF Page 0 Setting the Stage Transparency (x, y ) (x 1, y 1 )

26 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 1 Teacher Reference Activity 1 In this activity, students will write quadratic equations in standard and general form, and identify the coefficients a, b, and c. Students will also be introduced to the quadratic formula and the importance of the discriminant in determining the nature of the roots of the quadratic formula. Introduce the class to the standard form of a quadratic equation: y = ax + bx + c. Inform the class that the coefficients a, b, and c will play a very important part in today s lesson. Ask the students if they can determine what the y intercept will be for a quadratic equation written in standard form. We are gauging the student s prior knowledge of the y intercept to determine if they remember that the x-value is always 0 for the y intercept. Introduce the class to the general form of a quadratic equation: ax + bx + c = 0. Note: Many books and resources have different ways to write the standard form and the general form. We will use the equations stated above for our purposes. A quadratic function in standard form is written as f ( x) = ax + bx+ c. Ask the class what is the difference between the standard form and general form of a quadratic equation. The students need to recognize that y is set equal to zero for the general form. If the class realizes that y = 0, then ask the class If y = 0 and we solve the quadratic equation in the general form for x, what do we get? The students should realize that we get the solutions and the x intercepts of the quadratic equation. Another name that they may not be aware of is called the zeros of the quadratic equation. They are called the zeros when substituted for x; because, the quadratic equation evaluates to zero. Hence, solving a quadratic equation gives solutions, x intercepts, and the zeros. NOTE: The concept that we have three different names when solving a quadratic equation can be expanded for all polynomials of degree higher than. b± b 4ac Next, introduce the students to the quadratic formula, x =, for solving quadratic equations (y = a ax + bx +c) when y=0. Let the class know that this formula always works when trying to find the solution for a quadratic equation. Point out to the class the importance of the portion under the radical sign, called the discriminant (b 4ac). Have the class get into groups of four. Ask the class What are the three possible values for the discriminant and how do they affect the solutions for x? You might want to give the class a hint and say that one of the possible values are positive numbers. Have the students work in their groups to discuss the answer to the question. You might want to give the students 3 sets of values for a, b, and c to help them determine that the three values we are looking for are positive, zero, and negative. Here are three sets of values for a, b, and c: Use: b 4ac 1. a = 1; b = ; c = 4. a = 1; b = ; c = 1 3. a = 1; b = ; c = 3 The class should discover the following: 1. When the discriminant is positive (greater than 0) then these are two roots that give two distinct solutions for x.. When the discriminant is zero then there are two roots that are exactly the same that give a solution for x, which we call a double root.

27 AIIF Page 3. When the discriminant is negative (less than 0) then there are no real number solutions for x. Bring the class back together and have groups share their findings with the rest of the class. Have a volunteer record the unique responses on the board or on a blank transparency on the overhead projector. After all the groups have shared their findings, have the class agree on a set of findings that best describe the characteristics of the nature of the roots for the discriminant. Tell the class that for any given quadratic equation there is either no solution or two solutions. The two solutions could be either a double-root or two different roots. Now, use a strategy of your choice to model how to use the quadratic formula to solve a quadratic equation. Have the class follow along with you on the dry erase boards with the problems you model or have them do a different problem. Make sure to first write the quadratic equations in general form and identify the three coefficients a, b, and c. It is very important that students establish a process on how to properly use the quadratic formula by first writing the quadratic equation in general form and then identifying the coefficients that will be used in the formula. Students may just want to substitute the coefficients in the formula without using the correct procedure and then they will not understand how they could arrive at the wrong solutions. Here are a few examples to model: y = x + x 3 y = x + 3x 4 y = x + 5x y x x= 6 y 4x 1= 4x Have the class work in their groups on Exercises 1 through 6. Tell the students that for Exercise 6 they are to identify the values of a, b, and c from the given quadratic formula and to write the equation for these values. Bring the class back together and have student volunteers share their results with the rest of the class. You might want to finish the activity by leading a discussion of the important concepts learned during the activity. Have a volunteer record the class responses on the board or on the overhead transparency. Have the class record the responses in their student journals.

28 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 3 Activity 1 SJ Page 8 In this activity, you will be solving quadratic equations using the quadratic formula to find the values of x when y=0. Make sure the equations are written in general form before determining the coefficients a, b, and c. Standard Form: General Form: y = ax + bx + c ax bx c + + = 0 For the following exercises: a. Write the quadratic equation in general form. b. Identify the values of a, b, and c. c. State the nature of the roots by calculating the discriminant. d. Find all solutions, if any, for the quadratic equation. Quadratic Formula: ± x = b b 4ac a y = x + 3x 4 a x + 3x 4 = 0. b. a = 1; b = 3; c = 4. c. The discriminant is 5 which is greater than 0; there will be two real distinct roots. d. The solutions are x = 1 and x = 4. y = x + x+ 5 a x + 5x + = 0. b. a = ; b = 5; c =. c. The discriminant is 9 which is greater than 0; there will be two real distinct roots. d. The solutions are x = and x = 1/. y = x x a 3x 8x 3 = 0. b. a = 3; b = 8; c = 3. c. The discriminant is 100 which is greater than 0; there will be two real distinct roots. d. The solutions are x = 3 and x = 1/3. y x x + 4 = 5 a x + 4x 5 = 0 or x 4x + 5 = 0. b. a = 1 (or 1); b = 4 (or 4); c = 5 (or 5). c. The discriminant is 4 which is less than 0; there are no real roots. d. The are no real solutions. y + 6x 9= x a x 6x+ 9= 0. b. a = 1; b = 6; c = 9. c. The discriminant is 0; there will be one double root. d. The double root solution is x = 3.

29 AIIF Page 4 SJ Page 9 6. For the given quadratic formula, identify the values a, b, and c and write the matching quadratic equation in standard from. Note: use y = ax + bx + c. a. 5± 5 4(1)(3) x = (1) The values are a = 1, b = 5, and c = 3. The quadratic equation is y = x + 5x + 3. b. 7 ± ( 7) 4(3)( 4) x = (3) The values are a = 3, b = 7, and c = 4. The quadratic equation is y = 3x 7x 4 c. 8 ± ( 8) 4( 9)(1) x = ( 9) The values are a = 9, b = 8, and c = 1. The quadratic equation is y = 9x 8x + 1

30 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 5 Teacher Reference Activity In this activity, students will write a program for the quadratic formula. Ask the class what was the Setting the Stage about. Now ask the class if the graphing calculator is somewhat like a computer and could we write a program for our quadratic formula? Tell the class that there are actually many programs written for the graphing calculator and yes there are even games for the graphing calculator. If you haven t taken the time to show the class how to write a program for the classroom graphing calculator, you may want to do so now -- not only to show the class the programming keys on the graphing calculator but also the programming logic as well. This can be considered an activity for the whole class. To create and name a program, press the following key sequence Í. The following screen shots show the steps to create and name a program using the TI-83 or 84 Plus graphing calculator. Enter the name of the program as displayed in the screen shots on the next page, QUADRTIC, or some other name you prefer. There are three menus that are used for writing a program: CTL (program control and logic), I/O (input/output), and EXEC (execute an existing program; we'll not use this one). The relational and logic operators are under the TEST button (y ). The CTL menu consists of the following (only the commands used most often are listed below): 1:If Creates a conditional test. :Then Executes commands when If is true. 3:Else Executes commands when If is false. 4:For( Creates an incrementing loop. 5:While Creates a conditional loop. 6:Repeat Creates a conditional loop. 7:End Signifies the end of a block. Used with If Then, Else, For, While, and Repeat 8:Pause Pauses program execution. 9:Lbl Defines a label. 0:Goto Goes to a label. E:Return Returns from a subroutine. Within the main program, Return stops execution and returns to the home screen. F:Stop Stops execution. If you have CtlgHelp application, run it to have catalogue help available. Catalogue help will give you the format of any command or operator by pressing the à key.

31 AIIF Page 6 The I/O menu consists of the following (only the commands used most often are listed below): 1:Input Enters a value. :Prompt Prompts for entry of variable values. 3:Disp Displays text, value, or the home screen. 4:DispGraph Displays the current graph. 5:DispTable Displays the current table. 6:Output Displays text at a specified position. 8:ClrHome Clears the display. Make sure the class takes into consideration the discriminant when writing the program. The purpose of this activity is to show the class the power of the graphing calculator. Adjust the program as necessary for the classroom graphing calculator. You may want to skip the first two screen shots and start with the third, which displays a message about entering the values for a, b, and c. Note: View the screen shots below carefully. Some of the screen shots have the last line repeated in the next shot. This is to show connectivity because some of the commands can't be totally displayed and to help determine what was the last command entered. Have the students continue working in their groups to write the program. Suggest naming the program QUADRTIC or something similar. It is probably best that the class test their programs on the exercises from Activity 1 or or some quadratic equations you may have in a textbook. Bring the class together and have volunteers share their program with the class, using the graphing calculator view screen. Determine if the class enjoyed this activity. Tell the students that this experience can be useful in other courses such as physics or other mathematics courses.

32 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 7 Activity SJ Page 10 In this activity, your teacher will guide you through writing a program for the quadratic formula on the classroom graphing calculator. Test your program on the first activity. Things you will need to pay attention to in your program are: The discriminant Programming logic Data input Data output Calculations using the quadratic formula Use the supplemental exercises below to further test your program by solving for x when y=0. Round your answers to 3 decimal places. 1. y = x x The solutions are 3 and 1.. y = x + x 1 5 The solutions are approximately and y = x + x There are no real solutions. 4. y = x + x The solutions are approximately and y x x = The double root solution is Can you think of any improvements in the program you wrote? Answers may vary.

33 AIIF Page 8 Teacher Reference Activity 3 In this activity, students will continue using the quadratic formula to solve application problems. Discuss what the class might need to consider to solve quadratic equations that wasn t necessary in the first activity. The key idea here is that students will determine if any solution does not make sense in the real world application. For instance, if an answer has a negative value but that negative value does not make sense in the application problem, then the students will need to discard that particular answer. For example, in dropping a ball from a building it is determined that the ball hit the ground at times t = 6 and t = 3, then the 3 value must be discarded since time is not measured using negative numbers. Ask the class, "What is the height of the ball when it hits the ground?" Students should know that the height is zero, so to solve an equation involving something hitting the ground is the same as solving an equation in general form. Also, discuss with the class the concept of break even point. The break even point is where the revenue equals the costs. No money is made (positive profit) and no money is lost (negative profit). Other issues to discuss include: Farthest distance traveled involves the time it takes for something to hit the ground once it is thrown or shot skyward. Gravity plays an important roll in projectile motion (baseball thrown, model rocket launched, etc). If we have to find how long it takes until an object is a certain height, then we set the function representing the position to the given height. For example, if s(t) = 16t + 5t + 10 and we need to find the time that the object has a position, or height, of 50 feet, then the equation would be 50 = 16t + 5t The students would then have to rewrite the equation in general form to find t. Have the students continue working in their groups, but begin working with a partner, for Exercises 1 through 4. After each pair has completed the exercises, have the pair compare answers with the other pair in their group and settle any discrepancies in their answers either amongst themselves or with other groups. Have volunteers share their results with the class.

34 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 9 Activity 3 SJ Page 11 In this activity, you will continue to use the quadratic formula to solve quadratic equations for real-world applications. Use the same process from Activity 1 to find your solutions (write the equation in general form; identify the coefficients a, b, and c.) Make sure your answers make sense for the real-world application problem. Break Even Point The point where the revenue, R(x), equals the cost, C(x). Symbolically, R(x) = C(x). 1. You have a part time job working for a local machine shop. The owner plans to make a certain product to sell. The product's costs are related by the functioncx ( ) = x+ x and the owner knows he can sell the product for $35.00 each, giving him a total revenue of Rx ( ) = 35x, where x represents the number of items produced. The owner would like you to find the break-even points so he can determine the number of the product items he should produce each week. The break-even points are (5, 0) and (50, 0).. A ball is thrown downward from the top of a building into a river. The height of the ball from the river can be modeled by Ht ( ) = 16t 15t+ 600, where t is the time, in seconds, after the ball was thrown. How long after the ball is thrown is it 75 feet above the river? How long, to the nearest tenth of a second, does it take the ball to land in the river? It takes the ball 5.3 seconds until it is 75 feet above the river. It takes the ball about 5.7 seconds to land in the river. 3. It takes a 004 Corvette 4.3 seconds to accelerate from 0 to 60 miles per hour. The same car can do the quarter mile, 130 feet, in 1.7 seconds. The displacement function can be described by the equation st ( ) = 4.09t t. a. How far has the Corvette traveled after 4.3 seconds, to the nearest foot? The Corvette has traveled 99 feet in 4.3 seconds. b. How long does it take the Corvette to travel half a mile? Round your answer to the nearest tenth of a second. Note: A mile is 5,80 feet. It takes the Corvette approximately 19.8 seconds to travel half a mile. 4. The Coast Guard is testing two rescue flares from two competing companies. The Coast Guard plans to sign a contract with the company whose rescue flare travels the farthest. The Coast Guard fires the two flares into the air over the ocean. The paths of the flares are given by: Company A: y = x + x+ 15 x 56 Company B: y = + x where y is the height and x is the horizontal distance traveled. Determine which flare the Coast Guard should purchase by substituting y = 0 into each equation and finding x. What does the constant 15 represent in each equation? Company A s flare travels horizontally about 3968 feet, while company B s flare travels horizontally about 4537 feet. The Coast Guard should purchase the flare from company B. The constant 15 represents the height from which the flare was fired.

35 AIIF Page 30 SJ Page 1 Practice Exercises For Exercises 1 through 3: a. Write the quadratic equation in general form. b. Identify the values of a, b, and c. c. State the nature of the roots by calculating the discriminant. d. Find all solutions, if any, for x when y=0 for the quadratic equation. Round all answers to the nearest tenth. 1. y = x x a. 11x 10x 1 = 0 b. a = 11; b = 10; c = 1. c. The discriminant is 144 which is greater than 0; there will be two real distinct roots. d. The solutions are x = 1 and x = 1/11.. y = x + x a. 3x + 5x + 1 = 0 b. a = 3; b = 5; c = 1. c. The discriminant is 169 which is greater than 0; there will be two real distinct roots. d. The solutions are x = 3 and x = 4/3. 3. y = x x 8 8 a. x 8x 8 = 0 b. a = ; b = 8; c = 8. c. The discriminant is 0; there will be a double root. d. The solutions are x = and x =. 4. Cox s formula for measuring velocity of water draining from a reservoir through a horizontal pipe is 100HD = 4v + 5v, where v represents the velocity L H of the water in feet per second, D represents the diameter of the pipe in inches, H represents the height of the reservoir in feet, and L represents the length of pipe in feet. How fast is water flowing through a 30 foot long pipe with diameter of 4 inches that is draining from a pond with a depth of 30 feet? Round your answer to the nearest tenth of a foot per second. The velocity of the water in the pipe is approximately 84. feet per second. L D

36 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 31 SJ Page A ball is thrown upward with an initial velocity of 146 feet per second from a height of 7 feet. How long does it take the ball to hit the ground? The equation for projectile motion is s(t) = 16t +v 0 t + h 0, where s is the height of the projectile in feet, t is the time in seconds, v 0 is the initial velocity, and h 0 is the initial height. Round your answer to the nearest tenth of a second. It takes the ball 9. seconds to hit the ground. 6. For the given quadratic formula, identify the values a, b, and c and write the quadratic equation from these values. a. 11 ± (11) 4(5)(6) x = (5) The values are a = 5, b = 11, and c = 6. The quadratic equation is y = 5x + 11x + 6. b. 1 ± ( 1) 4( )( 19) x = ( ) The values are a =, b = 1, and c = 19. The quadratic equation is y = x 1x 19. c. For part b. above, will the quadratic equation have any real solutions? Explain. For part b. above there will not be any real solutions because the discriminant has a value of Find the mistake below and correct it. x 13x= 7 ( 13) ( 13) 4(1)(7) ± = (1) 13 ± = 13 ± 141 = 13 ± = 1.45 and = 0.55 The equation was not written in general form first. = x 13x 7 = 0 x 13x 7 ± = (1) 13 ± = 13 ± 197 = 13 ± = 13.5 and = 0.5 ( 13) ( 13) 4(1)( 7)

37 AIIF Page 3 SJ Page 14 Outcome Sentences I know that the discriminant portion of the quadratic formula is used to I know that the quadratic equation must be in form to be When solving real-world applications using the quadratic formula The part of the quadratic formula I don t understand is because

38 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 33 Teacher Reference Lesson Quiz Answers 1. a. x + 15x + 5 = 0 b. a = 1; b = 15; c = 5. c. The discriminant is 15 which is greater than 0; there will be two real distinct roots. d. The solutions are x 1.91 and x a. x 7x 13= 0 or x + 7x+ 13= 0 b. a = 1 or 1; b = 7 or 7; c = 13 or 13. c. The discriminant is 3 which is less than 0; there will be no real roots d. There are no real solutions

39 AIIF Page 34 Lesson Quiz Name: For problems 1 and : a. Write the quadratic equation in general form. b. Identify the values of a, b, and c. c. State the nature of the roots by calculating the discriminant. d. Find all solutions, if any, for x when y=0 for the quadratic equation. 1. = y x x. = 7 13 y x x

40 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 35 Lesson 3: Graphing Quadratic Functions and Their Applications Objectives Students will understand that functions are used to model and analyze real-world applications and quantitative relationships. Students will understand that functions come in many different forms and are often needed to solve or simplify abstract ideas. Students will see how to use technology to understand, simplify, and solve complicated abstract ideas. Essential Questions How can quadratic functions and applications of quadratic functions be graphed? Tools Student Journal Setting the Stage transparencies Dry erase boards, markers, erasers Graphing calculators Warm Up Problems of the Day Number of Days 3 days Vocabulary Symmetry Vertical line symmetry y-coordinate Seven-pin polygon Vertical translation Vertex Quadratic regression Horizontal translation x-coordinate Congruent

41 AIIF Page 36 Teacher Reference Setting the Stage Before placing the Setting the Stage transparency 1 on the overhead projector, lead a discussion with the class about symmetry. Ask the class, What does it mean if an object has symmetry? Have a volunteer list the class responses on the board or on the overhead projector. After students have given several responses, ask, What types of symmetry are there? Have the same volunteer list the class responses, but have the responding students come to the board to demonstrate the type of symmetry they suggested. Make sure the class agrees with the demonstration before moving on to another type. Now place the Setting the Stage transparency 1 on the overhead projector. Tell the class that the type of symmetry they will be doing is vertical line symmetry. Remind the class that line symmetry means that an object can be folded, or reflected, so that the two parts are congruent. Vertical line symmetry means folding or reflecting the object about a vertical line. Have the class work in groups of four. Ask the class which letters of the alphabet have vertical line symmetry. You might want to use the letter A as an example by drawing a vertical line down the middle of the letter. Tell the students they can either visualize the symmetry or fold the letter in half to determine vertical line symmetry. Give the groups a couple of minutes to determine which letters have vertical line symmetry. Have each group give a letter that has vertical line symmetry while a volunteer circles the given letters on the overhead transparency. Ask the class if all letters that have vertical line symmetry have been circled. If they haven t, ask groups to name the remaining letters. Next, place the Setting the Stage transparency on the overhead projector. Tell the class that the seven dots are arranged in a hexagonal pattern. Tell the class that a seven pin polygon is a closed shape made by joining the pins, or dots, with straight lines. Draw the following polygons on the board. Tell the students that their objective is to draw as many seven pin polygons with vertical line symmetry as they can. Give the class about two or three minutes to complete the activity. Have someone from each group draw a seven pin polygon with vertical line symmetry on the transparency. Here are some examples of seven pin polygons that have vertical line symmetry:

42 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 37 Setting the Stage Transparency 1 A B C D E F G H I J K L M N O P Q R S T U V X Y Z

43 AIIF Page 38 Setting the Stage Transparency Seven Pin Polygons

44 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 39 Teacher Reference Activity 1 Place Activity 1 transparency 1 on the overhead projector. Have the class draw the plotted points and corresponding curved graph on their dry erase boards as best they can. Tell the class that they can approximate the points. Have a volunteer model vertical line symmetry while the class uses dry erase boards and assists the volunteer as needed. Tell the students to use their knowledge of vertical line symmetry to reflect their plotted points across the y axis and then draw the corresponding curved graph. Lead a discussion with the class about the vertical line symmetry they did for the Setting the Stage and how it can be applied to a set of points and corresponding graph. Have the students hold up their dry erase boards and visually inspect their results. Ask, Does the graph have a minimum or maximum value? Explain why. Also, what is the minimum or maximum value? The class should determine the answer by how the graph opens. Remember a graph that opens upward has a minimum value and a graph that opens downward has a maximum value. Place Activity 1 transparency on the overhead projector with the bottom table covered. Tell the class to plot the points from the table on their dry erase boards while a volunteer plots the points on the board. Lead a class discussion about the vertical line of symmetry at x = 4 and how it might affect them in determining the reflected points. You also might want to ask the class which point was not reflected about the vertical line and why. Have the class draw the graph of the plotted points. Ask the class if they recognize the graph. The class should notice the shape of the graph from the quadratic functions they graphed in Lesson 1. You can either discuss or point out to the class that x values that are the same distance from the minimum value have the same y value. This is a very important concept for vertical line symmetry for quadratic functions. Ask the class Without plotting the points and drawing the graph, is there a way to determine if we have a minimum or maximum value? Explain why. Also, what is the minimum or maximum value? Students should base their answers on the y values. If the y values are decreasing and then increasing, the graph opens upward. Likewise, if the y values are increasing then decreasing, the graph opens downward. Uncover the bottom table on Activity 1 transparency. Continue the discussion about the point in the table that won t be reflected about an axis. Have a third volunteer plot the table values on the board and reflected points while the class plots and reflects on their dry erase boards. Ask the class what is different about this graph compared to the others they have done. Place Activity 1 transparency 3 on the overhead projector. Have volunteers state the equation of the vertical line of symmetry for each graph and table. Have the class agree on these equations. Note: These are four separate problems. Ask the students how they determined the equation for the vertical line of symmetry. Also, ask the students, Do the graphs or data tables have a minimum or maximum value? Explain why. What is the minimum or maximum value of each graph and data table? and "How is the minimum or maximum value related to the vertical line of symmetry?"

45 AIIF Page 40 Some of the important concepts that the students should understand, either in this pre-activity modeling and discussion or by the time they have completed the exercises in Activity 1, are: Opposite x-values have the same y-value (for vertical line of symmetry being the y axis.) Opposite x-values are the same distance from the minimum or maximum x-value (for vertical line of symmetry not being the y axis.) The left side of the graph is a mirror image of the right side and vice versa. The x coordinate of the minimum or maximum point determines the equation for the vertical line of symmetry. A graph that opens upward has a minimum value and a graph that opens downward has a maximum value. Have the class work in pairs on Exercises 1 through 4. Have each pair get together with another pair after completing the exercises. Within these groups, have student pairs share their results. If there are any discrepancies in their results, have the group check with other groups about their results and agree on the final results for each exercise. Then have the groups share their findings.

46 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 41 Activity 1 Transparency 1 y x 1 3

47 AIIF Page 4 Activity 1 Transparency Vertical Line of Symmetry: x = 4 x y Vertical Line of Symmetry: x = x y

48 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 43 Activity 1 Transparency 3 x y x y

49 AIIF Page 44 Activity 1 SJ Page 15 In this activity, you will use your knowledge of the vertical line of symmetry to plot points, draw a graph, and find the equation for the vertical line of symmetry. 1. Using the given dashed vertical line of symmetry, plot and draw the missing half of the graph. Write the equation for the vertical line of symmetry. State whether the graphs have a minimum or a maximum value and explain why. a. b. The equation for the vertical line of The equation for the vertical line of symmetry is x =. The graph opens x = 3. The graph opens downward so it has upward so it has a minimum. a maximum.. Complete the tables below using the values in the table along with the equation for the vertical line of symmetry. Plot the points in the table, draw the graph, and draw the vertical line of symmetry. State the equation of the vertical line of symmetry, whether the data tables have a minimum or a maximum value, and explain why. a. Vertical line of symmetry: x = 3. It has a maximum value. x y b. Vertical line of symmetry: x = 1/. It has a minimum value. x y 5 71/ 7/ 13 1/ 1/ 5 1 1/ 5/ /

50 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 45 SJ Page Write the equation for the vertical line of symmetry for the given graphs. State whether the graphs have a minimum or a maximum value and explain why. a. b. The equation for the vertical line of symmetry is x = 0. The graph opens up and has a minimum. The equation for the vertical line of symmetry is x = 9/ or 4.5. The graph opens down so it has a maximum. 4. Write the equation for the vertical line of symmetry for the data tables below. State whether the data tables have a minimum or a maximum value and explain why. Also state the minimum or maximum value. a. b. x y x y The equation for the vertical line of The equation for the vertical line of symmetry is x = 6. The y-values in symmetry is x = 6. The y-values in the data data table are decreasing and then the table are increasing and then decreasing increasing so there is minimum value so there is a maximum value of. of 8.

51 AIIF Page 46 Teacher Reference Activity In this activity, students will continue to explore the line of symmetry and will be introduced to the term vertex. Using a strategy of your choice, model how to find the equation of the line of symmetry for y = x 4x. Use the graphing calculator, view screen, and the techniques from the Lesson 1 to find the minimum value for the quadratic equation. Have the students write the equation for the line of symmetry on their dry erase boards and hold up the boards. Ask the class members what they wrote for the equation for the line of symmetry. Make sure the class agrees on this equation. Show the class how to use the line of symmetry to find values of x that have the same y value. Tell the class that the distance between the two x values that have the same y value is the same on each side of the vertical line of symmetry. This technique makes it easy to use symmetry to graph quadratic functions because we need to find only the minimum or maximum value and a few points on one side of the minimum or maximum point and their opposites by symmetry. For example, from the quadratic equation above, we found the minimum to be the point (1, 6). If we can find the y values for x equal to 0,, and 4, we could then use symmetry to get the same y values for the x values of, 4, and 6 (these three values of x are the same distance from the vertex as the first three x values). Tell the class that this minimum or maximum value has a special name, it is called the vertex. Ask if anyone has encountered this word before. They might have heard it as part of an angle or a regular polygon. Lead the following dialogue with the class: We saw in Lesson 1 that quadratic functions of the form y = ax, where a is any positive number, that the graph opened upward and the vertex point (a minimum) was the origin (0, 0). If a was a negative number for y = ax, the graph opened downward and the vertex point (a maximum) was the origin (0, 0). We also saw that if the quadratic function was of the form y = ax + k, where k is any positive number then the vertex point (minimum or maximum) was at (0, k). If the quadratic function was of the form y = ax k, where k is any positive number then the vertex point (minimum or maximum) was at (0, k). Likewise, quadratic functions of the form y = ax ( h ), where h is any positive number, have a minimum or maximum value of (h, 0). And, if the quadratic functions were of the form y = ax ( + h ), where h is any positive number, then the vertex point (minimum or maximum) was at ( h, 0). We know that h caused a horizontal translation of the vertex point while k caused a vertical translation. Using h and k together we can translate the vertex both horizontally and vertically at the same time. Now, what if we didn t have a graphing calculator and we had to find the minimum or maximum value to y = x 4x. How could we find the minimum or maximum value with out creating a table of values or a rough sketch of the graph? Are there any special formulas that could help us? It turns out there is such a b formula. The x coordinate of the vertex can be found by using: x = and the y coordinate can be found by a substituting the x value into the equation, or namely y = f( x ), where f(x) = ax + bx + c. Tell the class that the coordinates for the vertex are b b, f, but the equation must be written in the standard form a a y = ax + bx + c. Note that the number in front of the x squared term is a, the number in front of the x term is b, and the constant term is c. Does c help us find any particular point on the graph of a quadratic equation? The class should realize that a constant for any equation, linear or non linear, generally represents the y intercept.

52 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 47 Model, or have a volunteer model, finding the vertex for y = x 4x while the class parallels with y = x + 6x+ 8using dry erase boards, but not the graphing calculator. Note that a = 1, b = 4 and a = 1, b = 6, respectively. Using the formula for the vertex, the volunteer and the class should have calculated the vertex to be (, 6) and ( 3, 1), respectively. Have another volunteer model finding the vertex for the quadratic function y = 4x 8x + (x = 1 and y = ) on the board or overhead projector while the class finds the vertex for y = x 1x + 3 (x = 3 and y = 1). Model as many quadratic functions as needed by using the following examples. Examples y = x + x + 5, vertex is ( 1/, 19/4) or ( 0.5, 4.75). y = 4x 4x + 8, vertex is (3, 8). 1 3 y = x x +, vertex is (, 1/) or (, 0.5). y = 5x 15x 7, vertex is ( 3/, 17/4) or ( 1.5, 4.5). Have the class work in pairs on Exercises 1 through 5. Have volunteers share their results and the class agree on the results. If any students has a problem with any exercise, have another volunteer model his/her results on the board or overhead projector. Now ask the class, What are the techniques you can use to graph any quadratic function without a graphing calculator or other technological tool? Have a volunteer record the list of techniques, given by the class, either on the board or overhead projector. Each student should record this list in their student journals. Their list should include the following: Vertex y intercept x intercept(s) (discriminant) Line of symmetry The class should now be able to graph any quadratic function along with applications of quadratic functions. Using either the above list or the list the students compiled, model graphing the quadratic functions above while the students use their dry erase boards to graph the same equations. Have pairs of students get together to form groups of four for Exercises 6 and 7. Tell the class to use its understanding of quadratic functions to find the functions for the given graphs. You may want to model how to do this with Exercise 6. The students should be able to use their knowledge of the vertical line of symmetry, vertex, and the various forms of a quadratic function to do this. Bring the class together and have groups share their results on the board.

53 AIIF Page 48 SJ Page 17 Activity In this activity, you will be determining specific characteristics of quadratic functions and real world problems involving quadratic functions and then graphing the quadratic functions from the characteristics. In the following exercises you will need to: a. Identify the values of a, b, and c b. Vertex coordinates c. All intercepts d. Line of symmetry e. Several points on either side of the vertex b Vertex x coordinate: x = a b Vertex y coordinate: y = f a b Line of Symmetry: x = a Quadratic Formula: Discriminant: b 4ac ± x = b b 4ac a 1. y = x + 4x 5 a a = 1; b = 4; c = 5 b. Coordinates of the vertex are (, 1) c. The y-intercept is (0, 5); there are no x- intercepts d. Line of symmetry is x = e. Several points on each side of the vertex may vary. Sample points are ( 1, 10), (5, 10), ( 4, 37), and (8, 37). y = x x a a = 3; b = 8; c = 3 b. Coordinates of the vertex are (4/3, 5/3) c. The y-intercept is (0, 3); x-intercepts are (3, 0) and ( 1/3, 0) d. Line of symmetry is x =4/3 e. Several points on each side of the vertex may vary. Sample points are (, 7), (/3, 7), (, 5), and (14/3, 5) 3. y = x x+ 5 a a = ; b = 5; c = b. Coordinates of the vertex are (1.5, 1.15) c. The y-intercept is (0, ); x-intercepts are (0.5, 0) and (, 0) d. Line of symmetry is x =5/4 or x = 1.5 e. Several points on each side of the vertex may vary. Sample points are (0, ), ( 1, 9), (.5, ), and (3.5, 9)

54 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 49 SJ Page Photosynthesis is the process in which plants use the energy from the sun's rays to convert carbon dioxide to oxygen. The intensity of light is measured in lumens. Let R be the rate that a certain plant uses to convert the sun's light energy. Let x be the intensity of the light. The plant converts the carbon dioxide at a rate according to the equation R= 40x 80x. Sketch the graph of this equation and determine the intensity that gives the maximum rate of photosynthesis. State the domain which makes sense for the application. The intensity of light that gives maximum rate of photosynthesis is 1.5 lumens with a maximum rate of 180. The line of symmetry is x = 3/ (or x = 1.5); y- intercept is (0, 0); x-intercepts are (0, 0) and (3, 0). Some points on either side of the vertex are (1, 160), (, 160), (0.5, 100), and (.5, 100). Note: y-axis has a scale of The cost function to make a certain product is Cx ( ) = 0.x 10x The revenue function for the same product is given by Rx ( ) = 0.x + 50x. a. Graph the cost and revenue cost functions on the same set of axes. b. What level of production will produce the maximum revenue? What is the maximum revenue? The level of production that maximizes revenue is 15 units. The maximum revenue is $ c. What level of production will produce the minimum cost? What is the minimum cost? The level of production that minimizes cost is 5 units. The minimum cost is $ d. Graph the profit function (profit = revenue minus cost) on a separate set of axes. e. What level of production will produce the maximum profit? What is the maximum profit? The level of production that maximizes profit is 75 units. The maximum profit is $

55 AIIF Page 50 SJ Page The graph below represents the profit function for a company that produces widgets. Find the equation of the profit function P(x). Note: You should be able to determine the value of c from the graph. Also, use the coordinates of the vertex to find a and b. Use b/a for the x-coordinate and solve for b in terms of a and substitute this value into y = ax + bx + c to find a and then b. The value of a = -1, b = 100, and c = 100. From the graph we see that the y intercept is 100. Hence c = 100. To find "a" and "b", use vertex formula to get x = b/a or b = ax. Looking at the graph we see that the x-value of the vertex is 50 and the y-value is Using the equation 1300 = a(50) a(5) 100, we get that a = 1 and then b = 100. The quadratic equation of the graph is P(x) = x + 100x A town is having a parade and celebration for its high school marching band. The school s marching band recently marched in Macy s Annual Thanksgiving Day Parade. This was the first time the marching band is being honored for its hard work and achievement in the state competition. The town wants to hang a banner on a steel cable between its two tallest buildings -- each 100 feet tall. The distance between the two buildings is 50 feet. The weight of the banner caused the bottom of the banner to be 0 feet lower than the top of the building. Assume the bottom of the banner is parabolic in shape. What is the quadratic function that represents the lower portion of the banner? Answers may vary. A sample response might be: The quadratic equation that represents the bottom portion of the banner is y = 0.03x Draw a coordinate system such that the position of the origin is on the ground halfway between the buildings.

56 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 51 Teacher Reference Activity 3 In this activity, students complete quadratic data modeling by fitting data to quadratic equations. Remind the students that in the Linear Functions unit, they did linear data modeling. They drew the line of best fit and then determined the equation for the line they drew. Also, the last activity had the students determining quadratic functions from graphs of parabolas. Place the Activity 3 transparency on the overhead. Ask the class which scatter plots most resemble a parabola or have a quadratic trend. Students might not think that graph D on the right is a parabola, but if they look close and draw the parabola of best fit they could possibly see a portion of the arc of the parabola. Have the class answer Exercise 1 based on which scatter plots they felt had a quadratic trend. Have volunteers draw the best fit parabola on the graphs that most resemble a parabola; the rest of the class will work in pairs from the last activity to draw the best fit parabola so they can determine the quadratic function. Walk around the room and assist the class on Exercises and 3 as needed and give blank grid transparencies to pairs of your choice to have them share with the rest of the class. If necessary, review with the class how to enter data into the graphing calculator. The key sequence, for quadratic regression for the TI 83 or 84 Plus family of calculators, is ~. QuadReg is displayed to the home screen. By default, QuadReg uses L1 and L as the lists for x and y, respectively. If the data has been entered in different lists then press the necessary keys to get the appropriate lists. For example, if the lists for x and y were L3 and L4, respectively, then press y  (L3) y (L4). Pressing the Í key will execute the QuadReg command and display the results to the home screen. Don't worry about any diagnostics for the quadratic function; we won't be discussing diagnostics with quadratics as it is beyond the scope of this lesson. QuadReg requires at least three sets of ordered pairs. The following screen shots show selecting QuadReg, having it displayed to the home screen, and the results of executing the command. You also may opt to show the class, if you haven't before, how to do a scatter plot with their data sets. After the data have been entered using the key, press the y o keys (,) to display the STAT PLOTS menu. Select the desired stat plot (use the first one by default). Pressing the Í key to turn on the stat plot, select the desired Type: (first type is suggested), XList and YList default to L1 and L respectively. Select the Mark: the square mark is suggested. The screen shots below show the steps on doing a scatter (stat) plot.

57 AIIF Page 5 To have the best fit parabola sent to Y1 in the Y= editor to display the scatter plot and the best fit parabola at the same time, follow this key sequence: use the steps above to display QuadReg to the home screen; press ~ (to select Y VARS) Í (or À to select 1:Function ) À (or Í to select 1:Y1). Your home screen should have QuadReg Y1 displayed. Pressing the Í key will execute the command and the results will be displayed to the home screen and the equation to Y1 (see screen shots below). Have the class continue to work in their pairs on Exercises 4 through 6. Student pairs can check their results with one or more other pairs. Have volunteers share their results using the calculator view screen.

58 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 53 Activity 3 Transparency A B C D

59 AIIF Page 54 SJ Page 0 Activity 3 In this activity, you will use your knowledge and understanding of quadratic functions to do quadratic regression on scatter plots and data sets. In the last activity, you wrote quadratic functions from graphs. In the Linear Functions unit, you drew the best fit line for a scatter plot and determined the equation for the line of best fit. In this activity, you will use the concepts and skills developed in the Linear Functions unit to draw the best fit parabola for given graphs and then determine the equation for the parabola you drew. 1. Which scatter plots below seem to have a quadratic trend? Scatter plots A, C, and D seems to have a quadratic trend. A B C D. Draw a best fit parabola for the scatter plots you determined had a quadratic trend in Exercise 1. Answers will vary 3. Determine the quadratic functions from the best fit parabolas you drew in Exercise. Answers will vary

60 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 55 SJ Page 1 In the Linear Functions unit, you learned to use the graphing calculator to determine the equation for the best fit line from sets of data. We called this linear regression. The graphing calculator can also be used to determine the equation for the best fit parabola from sets of data. We call this quadratic regression. Follow your teacher's instructions on how to use the graphing calculator to determine the quadratic function from sets of data. 4. The table below shows the U. S. population distributed by age (x) and percentage (y). Under 5 5 to to to and over x y 7.4% 18.% 43.% 18.6% 1.6% a. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three decimal places. The equation for the best fit parabola is y = 5.943x x 4.84 b. Use your graphing calculator to create a scatter plot and graph of the data and sketch the scatter plot and graph on the set of axes. The graphing calculator plot and graph is: c. How well does the graph of the best fit parabola fit the data? Sample response: The graph of the best fit parabola does not reach the vertex, maximum value, of the data. 5. The students of Mr. G's class were told to record the number of hours spent studying for their mathematics test. For each student, Mr. G wrote an ordered pair (x, y). The x-value represented the number of hours the student spent studying and the y-value represented the student s test score. (0.5, 40), (9.3, 75), (8.4, 80), (0.5, 56), (1.0, 60), (8., 83), (7.6, 87), (1.0, 47), (1.4, 48), (7.0, 91), (6.5, 94), (1.5, 63), (.0, 73), (6. 98), (5.5, 100), (.3, 78),(.4, 83), (5.4, 97), (5.4, 98), (.5, 77), (.6, 83), (5., 95), (5.1, 85), (3.0, 88), (3.0, 86), (4.9, 94), (4., 93), (3.5, 91), (3.5, 90), (3.7,89). a. Use your graphing calculator to create a scatter plot. Does the data seem to model a quadratic equation? Explain. Sample response: Yes, the data seems to model a quadratic equation because the scatter plot is shaped like a parabola.

61 AIIF Page 56 SJ Page b. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three decimal places. The equation for the best fit parabola is y = 1.944x x c. Use your graphing calculator to create a scatter plot and graph of the data on the same set of axes. The graphing calculator plot and graph is: d. How well does the graph of the best fit parabola fit the data? The graph of the best fit parabola fits the data very closely. 6. The table below is the U. S. Census (in millions of people) for the years 1810 through 000. The x-values represent the year the Census was taken and the y-values represent the population in millions of people. Note: x = 0 for the year 1810, x = 10 for the year 180, etc x y x y a. Use your graphing calculator to create a scatter plot. Does the data seem to model a quadratic equation? Explain. Yes, the data seems to model a quadratic equation because the scatter plot is shaped like the right half of a parabola b. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three decimal places. The equation for the best fit parabola is y = 0.007x x c. Using your equation of best fit, predict the population for the Census in 010 and 00. The predicted population for 010 is million people and for 00 is million people

62 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 57 Practice Exercises SJ Page 3 Graph Exercises 1 and 3. Make sure to include the following: a. Identify the values of a, b, and c. b. Vertex coordinates. c. All intercepts. d. Line of symmetry. e. Several points on either side of the vertex. NOTE: Round answers to nearest tenth. 1. y = 5x 10x 1 a. a = 5; b = 10; c = 1 b. The coordinates of the vertex are (1, 6) c. The y-intercept is (0, 1); the x intercepts are approximately ( 0.1, 0) and (.1, 0) d. The equation for the line of symmetry is x = 1 e. Several points on either side of the vertex include: (0, 1), (, 1), (, 39), (4, 39), ( 4, 119), and (6, 119). y = x + x a. a = 3; b = 5; c = 1 b. The coordinates of the vertex are (5/6, 169/1) c. The y intercept is (0, 1); x intercepts are (3, 0) and ( 4/3, 0) d. The equation for the line of symmetry is x = 5/6 e. Several points on either side of the vertex include: (, 10), ( 1/3, 10), (5, 38), ( 10/3, 38), (8, 140), and ( 19/3, 140)

63 AIIF Page 58 SJ Page 4 3. A ball is thrown directly upward from an initial height of 00 feet with an initial velocity of 96 feet per second. After 3 seconds it will reach a maximum height of 344 feet. The standard form of a quadratic equation for a projectile is given by st () = 16t + vt 0 + s 0, where s(t) is the projectiles height at time t, v 0 is the initial velocity, and s 0 is the initial height. What is the equation of the quadratic function for this problem? What does the y intercept represent? Graph the quadratic function. Round answers to nearest tenth if necessary. The quadratic equation is s(t ) = 16t + 96t + 00 a. a = 16; b = 96; c = 00 b. The coordinates of the vertex are (3, 344) c. The y intercept is (0, 00); the x intercepts are approximately ( 1.6, 0) and (7.6, 0) d. The equation for the line of symmetry us x = 3 e. Several points on either side of the vertex include: (0, 00), (6, 00), (, 38), (4, 38), (7, 88), and ( 1, 88) f. The y intercept represents the initial height from where the ball was thrown NOTE: Vertical scale ratio is 1:14 4. Suppose that in a monopoly market (a market with a downward sloping curve) the total cost per week of producing a particular product is given by the cost function Cx ( ) = x + 100x The weekly demand for the product is such that the revenue function is Rx ( ) = x + 500x. Graph both functions on the same set of axes and shade the region that represents the area in which the company is making a profit. Find the points of intersection for the cost and revenue functions. What do the points of intersection represent? The points of intersection are (10, 4800) and (90, 8800). The points of intersection represent the break even points.

64 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page Determine the quadratic function from the graph at the right. The equation of the graph is y = x + 8x + 1. SJ Page 5 6. A ball was dropped from a height of approximately 5 feet and a motion detector was used to measure the time and height of the ball, relative to the ground, as it was falling. The table below is the height, h, of the ball off the ground in feet after t seconds. Time t Height h a. Determine the equation for the parabola of best fit. The equation for the best fit parabola is y = 13.57x 1.63x b. How long does it take for the ball to hit the ground? Round your answer to the nearest hundredth of a second. HINT: Use the quadratic formula. Sample response: Using the quadratic formula with a = 13.57, b = 1.63, and c = 4.949, the ball takes approximately 0.55 seconds to hit the ground 7. You run a bicycle rental business for tourists during the summer in your town. You charge $10 per bike and average 0 rentals a day. An industry 300 journal says that, for every 50 cent increase in rental price, the average 00 business can expect to lose two rentals a day. The graph to the right represents the quadratic equation 100 used to determine how many, if any, 50 cent increases are needed to maximize revenue. Let x represent the number of increases to the current charge rate. Negative values for x represent 50 cent decreases. Use this information and the graph to find the quadratic equation to maximize revenue. What should you charge per bike rental? What is your maximum profit? The quadratic equation for the graph is R(x) = x 10x+ 00. You should charge $7.50 per bicycle rental to have your maximum revenue of $5.

65 AIIF Page 60 SJ Page 6 Outcome Sentences The vertex is determined by The line of symmetry is used for Applications of quadratic equations really help me to understand Quadratic functions and applications of quadratic functions are graphed by The easiest way for me to write a quadratic equation from a graph is by The most difficult part of graphing is because Quadratic modeling with the graphing calculator

66 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 61 Teacher Reference Lesson 3 Quiz Answers 1. The hang time for the punt is approximately 5 seconds. The maximum height of the punt is 10 feet.. a. a = 1; b = 10; c = b. The coordinates of the vertex are (5, 3) c. The y intercept is (0, ); x intercepts are (3.7, 0) and (6.73, 0) d. The equation for the line of symmetry is x = 5 e. Several points on either side of the vertex include: (4, ), (6, ), (, 6), (8, 6), (0, ), and (10, ) 3. Answers will vary. A sample response might be: The best ways to communicate mathematical results for quadratic functions are through their maximum or minimum value, the vertex, as well as the zeros (also known as the solutions or x intercepts)

67 AIIF Page 6 Lesson 3 Quiz Name: 1. When a football player punts a football, he hopes for a long hang time. Hang time is the total amount of time the ball stays in the air. A time longer than 4.5 seconds is considered good. It allows the punting team time to get down the field and tackle the opponent s player who will catch the punt. If a punter kicks the ball with an upward velocity of 80 feet per second and his foot meets the ball feet off the ground, the function y = 16t + 80t + represents the height of the ball y in feet after t seconds. Sketch the graph of the punt. What is the maximum height the ball reaches? What is the hang time of the ball? Round your answer to the nearest tenth of a second and nearest tenth of a foot if necessary.. Graph y = x 10x+. Make sure to include the following: a. Identify the values of a, b, and c. b. Vertex coordinates. c. All intercepts. d. Line of symmetry. e. Several points on either side of the vertex. NOTE: Round answers to nearest hundredth. 3. What are the best ways to communicate mathematical results for quadratic functions in a meaningful manner?

68 Non Linear Functions Lesson 4: Power Functions AIIF Page 63 Lesson 4: Power Functions Objectives Students will understand what constitutes a power function. Students will be able to determine if a function is odd, even, or neither. Students will be able to graph power functions and their translations. Students will be able to solve and graph applications involving power functions Essential Questions Do power functions have patterns that can be used when solving and graphing them? Tools Student Journal Setting the Stage transparency Activity 1 transparency Dry erase boards, markers, erasers Graphing calculator and view screen Warm Up Problems of the Day Number of Days days Vocabulary Multiples Odd function Power function Patterns Even function

69 AIIF Page 64 Teacher Reference Setting the Stage Before placing the Setting the Stage transparency on the overhead projector, lead a discussion with the class about patterns. Ask the class about the geometry patterns they have learned about in school. They have also learned about patterns from graphs of equations and functions. Ask them about some numerical patterns they have learned about. Tell the class that in the Linear Functions unit they wrote rules for numerical patterns from tables. Let the class know that algebra looks at the numerical patterns more than the geometrical patterns. Lead a discussion about multiples. Tell the class that in elementary school they learned that repeated addition was called multiplication. The students should be familiar with multiples of numbers. Have several students give the multiples of different numbers such as, 3, 4, and 5. The class may have also seen that repeated multiplication results in a power (you may want to review the parts of a power, base and exponent.) Ask the class, "How are the multiples of a number related to the values of a power or exponentiation?" Place the Setting the Stage transparency on the overhead projector. Start with the number. Tell the class that the first multiple of any number is also the first power for that number. So, the first multiple of is also the first power of. The value of the power,, will tell us where the next multiple that has the same value as the next power. That is, the nd multiple has the same value of the next power which is 4. Note: The circles relate which multiples have the same value as the power of. The value of 4 tells us the 4 th multiple will be where the multiple values of equals the next power of, which is 8. The 8 th multiple of is where the next power of and multiple of have the same value, 16. The pattern continues on in this manner. The 16 th multiple is 3 which is also the value of the next power of ; 3 nd multiple of is 64 which is where the next power of is located. This 64 tells us the next multiple where the power of two has the same value as a multiple of two. Ask the class, "After 3, what is the next multiple of that has the same value as the next power of and what is that value?" Perhaps the class will say the 64 th multiple of, has the same value as the next power of which is 18. Now switch to the number 3 on the transparency. Again, remind the class that the first multiple of 3 will also be the first power of 3. That value, 3, tells us that the 3 rd multiple of 3 will be the next power of 3 which is 9. The value 9 tells us that the 9 th multiple of 3 has the same value as the next power of 3, 7. Have a student come to the overhead and complete the pattern for the number 3. Now ask the class to do a similar matching pattern for the number 4 on the blank side of their dry erase boards. Give the class five to seven minutes to complete the pattern matching for the number 4. Give the class graphing calculators if necessary. Have the students hold up their dry erase boards; walk around to visually inspect them while a volunteer shares her/his results on the transparency at the overhead.

70 Non Linear Functions Lesson 4: Power Functions AIIF Page 65 Setting the Stage Transparency

71 AIIF Page 66 Setting the Stage

72 Non Linear Functions Lesson 4: Power Functions AIIF Page 67 Teacher Reference Activity 1 In this activity, students will create, work with, and solve simple power functions and power equations. Think aloud as you model the following problem: The volume of a cube shaped box is 64 cubic inches. What are the dimensions of the box? Even though some students might be able to give the answer right away, make sure to model how to properly set up and solve these types of functions. Some of the concepts you should model are: Draw a sketch and label the unknown units. For example, because the length, width, and height of a cube are all equal, we could label each dimension with s. Write a function, using unknowns representing the problem then substitute the given information. For 3 example, the function would bevs () = sand substituting the volume of 64 would yield the power 3 equations 64 = s or s 3 = 64. Solve the equation. Make sure the solution includes appropriate units. For this example, the solution would be s = 4 inches. Have a student model the following problem, on the overhead projector, while the class members follow along on their dry erase boards. 4 A number raised to the fourth power is 81. ( n = 81 ) If the students come up with only one solution, ask them if there is any other number, when raised to the fourth power, which would equal 81. The goal is for the students to understand that two different numbers, when raised to the fourth power, would equal 81, namely 3 and 3. Now ask the class when there might be only one answer to a problem. The understanding here is that the students can differentiate between problems with just numbers and real-world applications and that sometimes answers just don t make sense. If you feel the students need to see another example, have another volunteer model the following problem on the overhead projector, while the class follows along on their dry erase boards. The sum of a number and three, raised to the fifth power is thirty two. This problem is a bit more complicated than the previous one. For the class to write the correct equation you may want to ask questions such as, What is being raised to the fifth power? What must be calculated first, 5 the sum or the power? These questions are to help the students get the equation ( x + 3) = 3. These equations are similar to those in Lesson 1 except they include powers other than squaring. These questions also help the teachers to see how the students will handle solving power equations of this type. It is helpful to 5 see the type of strategies the students use to solve an equation such as ( x + 3) = 3, in order to make adjustments in teaching strategies. Students should be using the same strategies that they have been using to solve equations in earlier lessons. Make sure the students used their previous equation solving skills and make any adjustments if necessary.

73 AIIF Page 68 Model the following application problem to the class. The class should use their dry erase boards to solve the same, or a similar, problem simultaneously. Place Activity 1 transparency on the overhead projector so that the students can follow along while you read the problem. Ask guiding questions such as, "Which number is used to replace P in the formula?" and "Which number is used to replace S?" If P is invested at an interest rate r per year, compounded annually, the future value S at the end of the nth year is P(1 + r) n. The function that models this is S= P(1 + r ) n. What interest rate must a person obtain for a $10,000 investment to have a future value of $16,88.95 after 10 years? You may want to model for the students how to calculate the tenth root using their graphing calculators. Let the students know that they still use the exponent key,, but the exponent for a root is the reciprocal of the original exponent, 1/10. Also inform the students that they should use parentheses for rational exponents because the calculator uses the order of operations. You could ask, "Is there an undoing operation that will allow us to undo the power of 10?" The goal is that students understand undoing the operation of a power. The problem looks like the following: = 10000(1 + r) = (1 + r) = (1 + r) 1.05 = 1 + r 0.05 = r 10 Solution: The interest rate for the investment should be 5%. Ask the students, "Should we consider the negative root, 1.05? Explain." The students should understand for this type of problem the negative root would not make sense because we cannot have a negative interest rate. Another method to determine an n th root with a graphing calculator is to press the key and then press the key to select the 5: x option. To use this option the students would enter the root index value first, which is 10 from the example above. Next press the key followed by the key to display x to the home screen. Next enter the value and press the Í key. The screen shots below display the process to use this technique for roots on the TI 83 or 84 Plus graphing calculator.

74 Non Linear Functions Lesson 4: Power Functions AIIF Page 69 Have the students work with their partner on Exercises 1 through 5. Have the students go through the exercises together and verify answers. When they have finished and are sure of their solutions, have them group with another pair to compare their answers. After the students have verified their answers with another pair, have student volunteers share their answers with the class. They can write their solution on the overhead or on the board. While the students are working, circulate to ask guiding questions and provide encouragement. You might want to model Exercise 5 with the class or have a student model this exercise.

75 AIIF Page 70 SJ Page 7 Activity 1 In this activity, you will write and solve power type equations and their applications. Let's look at the following application problem: The area of a cube shaped box is 64 cubic inches. What are the dimensions of the box? What are the steps necessary to setup and solve problems of this type? We need to start by labeling the known and unknown (variable) information. Next, we need to write an equation with a single variable from the given information. Then, we need to solve the equation and answer the original question or questions. For Exercises 1 through 3, write and solve a power type equation. 1. The cube of a number is 15. What is the number? The equation is n 3 = 15. Solving this equation we get that the number is 5; n = 5.. Six is added to a number that was raised to the sixth power. If the sum is 735, what was the number that was raised to the sixth power? The equation is n 6 + 6= 735. Solving this equation we get that the number is 3 or 3; n = 3 or n= The difference of a number and six, raised to the fourth power, is 56. What are the numbers? The power equation is ( n 6) 4 = 56. Solving this equation we get negative or positive 10: ( ) 4 4 = 4 n 6 56 n 6= ± 4 n=± 4+ 6 n= 4+ 6= 10 or n= 4+ 6 n= 4. The volume of a spherical weather balloon is 53.3 cubic meters. What is the diameter of the weather 4 3 balloon? NOTE: The formula for the volume of a sphere isv = π r where r is the radius. Use 3.14 for 3 the value of π. Solving this equation we get that the diameter of the weather balloon is 10 feet.

76 Non Linear Functions Lesson 4: Power Functions AIIF Page = (3.14)r = (3.14)r = 3.14r r = SJ Page 7 (cont.) = 3.14 r 5 r A couple plans to invest $5, into an account that is compounded annually for 5 years. They hope to have $75, after the 5 years. What interest rate will guarantee that their investment of $5, will grow to $75, after the 5 years? NOTE: S = P(1 + r) t, where S is the value of the investment, P is the amount invested, r is the interest rate (as a decimal), and t is the number of years invested. The interest rate they need is r = 4.5%. 5 75, = 5,000(1 + r ) 75, ,000 5 = (1+ r) 5,000 5, = (1 + r ) = (1 + r ) = 1 + r = r = r 5

77 AIIF Page 7 Activity 1 Transparency If P is invested at an interest rate r per year, compounded annually, the future value S at the end of the n th year is P(1 + r) n. The function that models this is S= P(1 + r) n. What interest rate must a person obtain for a $10,000 investment to have a future value of $16,88.95 after 10 years?

78 Non Linear Functions Lesson 4: Power Functions AIIF Page 73 Teacher Reference Activity In this activity, students will investigate the graphs of power functions using the classroom graphing calculator. Model how to graph y = x and y = x 4 at the same time on the graphing calculator while the students graph them on their calculators. The class should be familiar with the first equation. Use the TRACE key to switch between the graphs so the class can determine which graph represents which equation. You may want to have a list of questions prepared to ask the students after the graphs are displayed on the calculator view screen and on the students' calculators. Here are some questions: What is the shape of the graph? Does the graph have a minimum or maximum value? What do you notice about the portion of the graph to the left of the y axis compared to the portion to the right of the y axis? What are the similarities and differences between the two graphs? What might be the best way to determine which graph is which without using a table of values or a TRACE key? 4 6 The graphs below are of the functions y = x, y = x, and y = x The graphs below are of the functions y = x, y = x, and y = x. Have a student volunteer record the responses on the board while the students record the responses in their student journals.

79 AIIF Page 74 Next, have a volunteer model graphing the two equations y = x 3 and y = x 5 on a graphing calculator. Have the student, along with the class list the similarities and differences between these equations. Use the TRACE key to switch between the graphs so the class can determine which graph represents which equation. You may want to have questions prepared to ask after the graphs are displayed on the graphing calculator view screen and on the students' graphing calculators. Here are some questions: What is the shape of the graph? Does the graph have a minimum or maximum value? Is there symmetry of any type? What are the similarities and differences between the two graphs? What might be the best way to determine which graph is which without using a table of values or a TRACE key? Have the class complete Part A of Activity in their journals in their pairs. Have student pairs share their results with the class. Students could also investigate, using the TRACE key, multiplying by a coefficient other than 1 and comparing it to multiplying by a coefficient of 1. For example, Y1 could contain y = x 3 and Y could 3 contain y = 3x. Students could predict what they expect the results to be before graphing the equations or looking at the table of values for the equations. Now, have the class complete Part B of Activity in their journals in their pairs. Have student pairs share their results with the class. Power functions of the type y = x n, where n is an even integer greater than or equal to, are called even functions because the left half of the graph is a mirror image of the right half (vertical symmetry). They are also called even functions because you obtain the same y values for both positive and negative values of x. For example, if y = 4 when x = 1, then y = 4 when x = 1. In function notation, a function is an even function when f (x) = f ( x). Power functions of the type y = x n, where n is an odd integer greater than or equal to 3, are called odd functions because the left half of the graph is a mirror image of the right half only reflected about the x axis. They are also called odd functions because you obtain the opposite y values for negative values of x that you obtained for positive values of x. For example, if y = 4 when x = 1, then y = 4 when x = 1. In function notation, a function is an odd function when f (x) = f ( x). It is advised that students look at symmetry and/or table values to determine if the function is an even function or an odd function. Model using the graphing calculator how the students could enter the function into the Y= and then check the table values to determine when the function is even or odd. Students may think that they only have to look at the exponent to determine if the function is even or odd. Although even functions will have an even exponent and odd functions will have an odd exponent, if the function has any translation from the origin the function will no longer be an even or odd function since it will not adhere to the rules to be an even or odd function. Now, we will investigate horizontal and vertical translations for power functions. Have the class use their graphing calculators and input the equation y = x into Y1 (make sure to have them clear out any existing equations in the Y= editor.) Tell the class to add any number between 5 and +5 after x. Have students share their results and any conclusion from the number they added. You could have the students who are sharing their results, use the calculator view screen to display their graph while they share their results and any conclusions. Have the class work on Part C in Activity and have students share their findings with the class.

80 Non Linear Functions Lesson 4: Power Functions AIIF Page 75 Use the same TRACE process to have students add a number between 5 and +5 to the x value before squaring. For example, use ( x + ). You may want to show the class how to do this by pressing the following TI 83 or 84 Plus (please modify the steps for your graphing calculator) key sequence à Á (or any other number). Their equation should resemble something like y = ( x + ). Again, have students share their results and any conclusion from the number they added. You could have the students who are sharing their results, use the calculator view screen to display their graph. From the two previous investigations, the class should understand that adding a value after x results in a vertical translation and adding a value with x before squaring results in a horizontal translation. Ask the class, "What results would we expect if we put the two translations together? Can you give an example to support your explanation?" Students should be able to give examples of the two translations in a single equation, such as y = ( x + 3) +. Ask the class, "Would changing the exponent, from two to three, affect the translations? Explain." The class should realize that the translations would be unaffected by a change in exponents. Have the class work on Part D in Activity and have students share their findings with the class. Have the students work in pairs on Exercises 1 through 10. Have students share their results with the class. Have individuals or student pairs check their results with other students or student pairs. Lead a class discussion on the characteristics and graphs of power or power like functions. Include guiding questions such as: What are the major differences between odd and even functions? How can the differences between odd and even functions be used to identify their graphs? What could be the best method to determine the equation of a power function from its graph? How do translations affect the graph of power functions? How does multiplying a power function by a constant affect its graph? Note: For Practice Exercise 10, you may want to discuss or model this problem before assigning it to the class. Lead a discussion with the class about the techniques they could use to solve the problem. Have a student list the techniques on the board and the class could write these techniques and ideas down in their student journal by the exercise to have when they are trying to solve the problem.

81 AIIF Page 76 SJ Page 8 Activity In this activity, you will investigate the graphs of power and power like functions. In your descriptions include whether the graph is an even or odd function. Part A: How do different powers affect the graph of y = x? n Function n y = x y Describe or Draw General Shape 3 = x See Teacher Reference for the general shape of the graph Describe location of maximum or minimum No maximum or minimum Describe similarity or difference to the graph of y = x The graph is not similar at all to y = x. This is the graph of an odd function y 4 = x Parabolic in shape The minimum value is at (0, 0) The graph is very similar to the graph of y = x y 5 = x See Teacher Reference for the general shape of the graph No maximum or minimum The graph is not similar at all to y = x. This is the graph of an odd function y 6 = x Parabolic in shape The minimum value is at (0, 0) The graph is very similar to the graph of y = x y 7 = x See Teacher Reference for the general shape of the graph No maximum or minimum The graph is not similar at all to y = x. This is the graph of an odd function Write your overall conclusion as to how different powers affect the graph of y = x. The answers will vary. A sample response might be: "As the value of n increases the graph of y = x n becomes more condensed and compact. The shapes of even powers of x stay roughly the same and the shapes of odd powers of x stay roughly the same as well." n

82 Non Linear Functions Lesson 4: Power Functions AIIF Page 77 Part B: How do different coefficients affect the graph of y = ax? n SJ Page 9 y y y Function n y = ax Describe or Draw General Shape = x Parabolic in shape but reflected about the x axis = 4x Parabolic in shape but grows at a faster rate than y = x = 9x Parabolic in shape but y = y y y 1 x reflected about the x axis Parabolic in shape but reflected about the x axis and decreasing at a slower rate than y = x 3 = x Same shape as y = x 3 but reflected about the x axis 3 = 5x Same shape as y = x 3 but reflected about the x axis and decreasing at a faster rate than y = x 3 3 = 5x Same shape as y = x 3 but increasing at a faster rate than y = x 3 Describe location of maximum or minimum Maximum value at (0, 0) Minimum value at (0, 0) Maximum value at (0, 0) Maximum value at (0, 0) No maximum or minimum No maximum or minimum No maximum or minimum Describe similarity or difference to the graphs 3 of y = x or y = x The graph is similar to the graph of y = x but is reflected about the x axis The graph is similar to the graph of y = x but increases at a faster rate The graph is reflected about the x axis and decreases at a faster rate The graph is reflected about the x axis and decreases at a slower rate The graph is similar to the graph of y = x 3 but is reflected about the y axis The graph is reflected about the y axis and decreases at a faster rate The graph is similar to the graph of y = x 3 but increases at a faster rate y = 1 x 3 Same shape as y = x 3 but increasing at a slower rate than y = x 3 No maximum or minimum The graph is similar to the graph of y = x 3 but increases at a slower rate Write your overall conclusion as to how different coefficients affect the graph of y = ax. The answers will vary. A sample response might be: "A coefficient greater than 1 causes the graph to increase at a faster rate. A negative coefficient causes the graph to be reflected about the x axis." n

83 AIIF Page 78 SJ Page 30 n Part C: How does adding or subtracting a constant, k, to y = x affect the graph of the equation? y y y y y y y Function n y = x ± k Describe or Draw General Shape = x + 1 Parabolic in shape but vertically translated by 1 unit from the origin = x + 3 Parabolic in shape but vertically translated by 3 units from the origin = x Parabolic in shape but vertically translated by units from the origin = x 4 Parabolic in shape but vertically translated by 4 units from the origin 3 = x + 5 Same shape as y = x 3 but vertically translated by 5 units from the origin 3 = x + 7 Same shape as y = x 3 but vertically translated by 7 units from the origin 3 = x 6 Same shape as y = x 3 but vertically translated by 6 units from the origin Describe location of maximum or minimum Minimum value at (0, 1) Minimum value at (0, 3) Minimum value at (0, ) Minimum value at (0, 4) No maximum or minimum No maximum or minimum No maximum or minimum Describe similarity or difference to the graphs 3 of y = x or y = x The graph is similar to the graph of y = x but is vertically translated from the origin The graph is similar to the graph of y = x but is vertically translated from the origin The graph is similar to the graph of y = x but is vertically translated from the origin The graph is similar to the graph of y = x but is vertically translated from the origin The graph is similar to the graph of y = x 3 but is vertically translated from the origin The graph is similar to the graph of y = x 3 but is vertically translated from the origin The graph is similar to the graph of y = x 3 but is vertically translated from the origin Write your overall conclusion as to how adding or subtracting different constants affect the graph of y = x. The answers will vary. A sample response might be: "The graphs are the same as y = x or y = x 3 but the graphs have been vertically translated." n

84 Non Linear Functions Lesson 4: Power Functions AIIF Page 79 SJ Page 31 Part D: How does adding or subtracting a constant, h, to the x value before completing the power, in the n equation y = x, affect the graph? y Function y = ( x± h) n Describe or Draw General Shape = ( x+ 1) Parabolic in shape but horizontally translated by 1 unit from the origin Describe location of maximum or minimum Minimum value at ( 1, 0) Describe similarity or difference to the graphs 3 of y = x or y = x The graph is similar to the graph of y = x but is horizontally translated from the origin ( x 3 ) y = + Parabolic in shape but horizontally translated by 3 units from the origin Minimum value at ( 3, 0) The graph is similar to the graph of y = x but is horizontally translated from the origin ( x ) y = Parabolic in shape but horizontally translated by units from the origin Minimum value at (, 0) The graph is similar to the graph of y = x but is horizontally translated from the origin ( x 4 ) 3 y = + Same shape as y = x 3 but horizontally translated by 4 units from the origin No maximum or minimum The graph is similar to the graph of y = x 3 but is horizontally translated from the origin ( x 6 ) 3 y = + Same shape as y = x 3 but horizontally translated by 6 units from the origin No maximum or minimum The graph is similar to the graph of y = x 3 but is horizontally translated from the origin ( x 3 ) 3 y = Same shape as y = x 3 but horizontally translated by 3 units from the origin No maximum or minimum The graph is similar to the graph of y = x 3 but is horizontally translated from the origin Write your overall conclusion of how adding or subtracting a constant to the x value in the equation y = x affected the graph. The answers will vary. A sample response might be: "The graphs are the same as y = x or y = x 3 but the graphs have horizontally translated." Write your overall conclusion as to what affect the values of a, h, k, and n have on the graph of y = ax ( ± h) n ± k. The answers will vary. n

85 AIIF Page 80 SJ Page 3 For Exercises 1 through 4, determine the following: a. Determine if the graph represents a power or power like function or not. b. Determine if the graph has a maximum or minimum value. If it does, state the value of the maximum or minimum. c. If the function represented by the graph is a power function, determine if it is even, odd, or neither. 1.. a. It is a power like function a. Is a power like function b. Has a minimum value of 9 b. Has a maximum value of 8 c. The function is neither even nor odd c. The function is neither even nor odd a. Is not a power function a. Is a power function b. Seems to have a minimum value b. There is no maximum nor approaching 0 minimum c. This is neither an even nor odd function c. It is an odd function

86 Non Linear Functions Lesson 4: Power Functions AIIF Page 81 SJ Page Match the equation with its graph. a. y = x 4 This equation matches graph B. b. y = x 3 This equation matches graph C. c. y = x 3 This equation matches graph A. A. B. C.

87 AIIF Page 8 SJ Page 34 For Exercises 6 through 10, state any vertical or horizontal translation from the first equation to the second. Sketch a rough graph of the equations showing translation (do not worry about scale). 6. y = x 3 and y = x There is a vertical translation of +4 units from the origin. 7. y = x 4 and y = x 4 3. There is a vertical translation of 3 units from the origin. 8. y = x 5 and y = (x 6) 5. There is a horizontal translation of +6 units from the origin. 9. y = x 5 and y = (x 6) 5. There is a vertical translation of units and a horizontal translation of +6 units from the origin. 10. y = x 6 and y = (x + 1) There is a vertical translation of +5 from the origin and a horizontal translation of 1 from the origin.

88 Non Linear Functions Lesson 4: Power Functions AIIF Page 83 SJ Page 35 Practice Exercises For the Exercises 1 through 3, write and solve a power like equation. 1. The fifth power of a number is 43. What is the number? The power equation is n 5 = 43. Solving this equation we get the number 3; n = 3.. Nine is subtracted from a number that is raised to the seventh power. If the difference is 119, what was the number that was raised to the seventh power? The power equation is n 7 9 = 119. Solving this equation we get the number ; n =. 3. The sum of a number and three, raised to the third power, is 1,331. What is the number? The power equation is (n + 3) 3 = Solving this equation we get 8: ( ) = 3 n n+ 3= 11 n+ 3 3= 11 3 n= 8 4. The volume of a cubic box is approximately 151 cubic inches. What are the lengths of the sides of the cubic box? Round your answer to the nearest tenth of an inch. The volume formula for a cube box is V = s 3. Solving this equation we get that the length of the sides of the box is approximately 11.5 inches. 5. Darnell and Shanice plan to invest $50, into an account that is compounded annually at a rate of 3.5%. Create a table of values that represents what their investment is worth after 4, 8, 1, and 16 years. NOTE: S = P(1 + r) t, where S is the value of the investment, P is the amount invested, r is the interest rate (as a decimal), and t is the number of years invested. Round the value of the investment to the nearest cent. Years Invested (t) Value of Investment in dollars (S) 4 57, , , , Darnell and Shanice plan to use the total value of the investment in 16 years for a college education for their only child. Approximately how much will they have available each year, for four years, for their child's education? Round your answer to the nearest thousand dollars. The couple will have about $1,675 per year for their child's college education.

89 AIIF Page 84 SJ Page 36 For Exercises 7 through 9, complete the following: a. State if it is a power or power like function or not. b. State if it has a maximum or minimum value and state the value of the maximum or minimum. c. State if it the function is even, odd, or neither. d. State any vertical or horizontal translation from the origin. e. Sketch a rough graph of the power or power like function. 7. y = x 4 a. The equation is a power function. b. The function has a maximum value of 0 at x = 0. c. The function is an even function. d. There are no translations from the origin. 8. y = (x + 3) 3 a. The equation is a power like function. b. The function has no maximum or minimum values. c. The function is an odd function. d. There is a horizontal translation of 3 from the origin. 9. y = (x ) 5 4 a. The equation is a power like function. b. The function has no maximum or minimum values. c. The function is an odd function. d. There is a horizontal translation of + and a vertical translation of 4 from the origin.

90 Non Linear Functions Lesson 4: Power Functions AIIF Page 85 SJ Page Determine the power like function from the given graph. The equation for the power function is y = (x ) 3. 5 (5, 7) (, 0) ( 1, 7) 5

91 AIIF Page 86 SJ Page 38 Outcome Sentences A power function is The difference between an even and an odd function is Applications of power functions really help me to understand When graphing power functions Vertical and horizontal translations from the origin are The most difficult part of power functions is

92 Non Linear Functions Lesson 4: Power Functions AIIF Page 87 Teacher Reference Lesson 4 Quiz Answers 1a. n 4 = 56; the number is 4 or 4. 1b. (n 3) 5 = 43; the number is 6.. The power like function has a horizontal translation of 1 unit from the origin and a vertical translation of units from the origin. The coordinates of the minimum value are (1, ). 3. a. The graph represents a power like function. b. The graph has no minimum or maximum value. c. The function is neither even or odd because it has been translated horizontally and vertically from the origin. d. There is a horizontal translation of units from the origin and a vertical translation of a 3 units from the origin. e. The equation is y= ( x+ ) 3 3

93 AIIF Page 88 Lesson 4 Quiz Name: 1. Write and solve a power type equation for the following: a. A number raised to the fourth power is 56. What is the number? b. The difference of a number and 3 raised to the fifth power is 43. What is the number? y = x 1 +, state any vertical and/or horizontal translation from the origin, state the coordinates of the minimum or maximum value if there is one, and sketch a rough graph of the power like function.. For the given power like function, ( ) 4 3. For the given graph: a. State if the graph represents a power or power like function or not. b. Determine if the graph has a maximum or minimum value and state the value of the maximum or minimum. c. Determine if the graph represents an even function, odd function, or neither. d. State any vertical or horizontal translation from the origin. e. Write the equation for the graph Note: Units on x axis are scaled 1:1

94 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 89 Lesson 5: Inverse Variation Objectives Students will be able to write equations involving direct variation applications Students will be able to calculate the constant of proportionality k Students will be able to write equations involving inverse variation applications Students will be able to graph direct variation equations Students will be able to graph inverse variation equations Students will be able to identify inverse variation phrases Essential Questions How is inverse variation used in real world application problems? How is direction variation used in real world application problems? Tools Student Journal Setting the Stage transparency Dry erase boards, markers, erasers Graphing calculator and view screen Construction paper Warm Up Problems of the Day Number of Days days Vocabulary Direct Inverse variation Inverse Constant of proportionality Direct variation

95 AIIF Page 90 Teacher Reference Setting the Stage Place the Setting the Stage transparency on the overhead projector. Cover the bottom part displaying the first two lines of text containing the phrases "Direct Variation Inverse Variation", and "Varies Directly Varies Inversely." Lead a discussion about Direct and Inverse. Ask the class, What is the difference between direct and inverse? Can you give examples of each? Have the class work in groups of four and give them minutes to come up with a list of differences between direct and inverse. Have groups share their list with the rest of the class. Have a volunteer list the class responses on the board or on the overhead projector. Now uncover the rest of the Setting the Stage transparency. Have volunteers read each statement. Have the students continue working in their groups and have them discuss the similarities and differences between the statements. Tell the students to determine the two variables in each statement and which one is the independent variable and which is the dependent variable. This will get the students to think about their understanding of variables discussed in the Linear Functions unit. Have different groups share their results for at least one of the statements. The key concept is that the students recognize that for inverse variation, the dependent variable decreases proportionately as the independent variable increases.

96 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 91 Setting the Stage Transparency Direct Variation Varies Directly The number of cricket chirps increases as the temperature increases. The amount of money earned increases as the number of hours worked increases. School grades increase as the number of hours spent studying increases. Miles driven increase as the time spent driving increases. Amount of confidence increases as our grades increase. Inverse Variation Varies Inversely The amount of gasoline decreases as the miles driven increases. The temperature of hot cocoa decreases as the amount of time increases. The amount of available cell phone minutes decreases as the amount of time we use our cell phone increases. The amount of available energy we have decreases as the amount of time exercising increases. Amount of available money decreases as the number of items purchased increases.

97 AIIF Page 9 Teacher Reference Activity 1 In this activity, students will create a bar graph that represents the equation y = 1/x, for x values 1 through 10. Have the students work individually or in pairs. Tell the class to cut out the grid template and the strip cutouts. Point out the 1 unit location on the vertical axis on the grid. The 1 unit value represents the length of one of the strips. Have the class measure one of the strips in millimeters. The 1 unit value represents 180 millimeters (mm). Tell the class to place the 1 unit strip above the 1 on the horizontal axis. Have the class cut the remaining strips so that their lengths represent the fractions 1/, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10 and place them along the horizontal axis above the values for through 10. For 1/7, tell the class to round to the nearest millimeter, 6. For 1/8, tell the class to try to cut half way between and 3 millimeters to get the.5 length. Provide each student or student pair with scotch tape to tape down each strip. Have the class write the fraction values above each strip on the grid and then calculate the decimal value for the fraction to the nearest hundredth. Students should also record these fraction and decimal values in the table provided in their student journal. A sample of what the bar graph should look like is displayed below. y 1 unit 1 1/ 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/ x

98 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 93 Activity 1 SJ Page 39 In this activity, you will create a bar graph that represents the equation y = 1/x, for x values 1 through Cut out the grid template. Obtain a piece of construction paper from your teacher and cut a strip that is 1 centimeter wide by 180 millimeters long. Notice the 1 unit location on the vertical axis of the grid. The 1 unit value represents the length of one of the strip cut to 180 millimeters.. Place the 1 unit strip at unit 1 on the x axis. Cut the remaining strips so that their lengths represent the fractions 1/, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10 the length of the unit strip cut in Exercise 1. Place the cut strips along the horizontal axis at the values for through 10. Tape down each strip. 3. Write the fraction values above each strip on the grid and then calculate their decimal values to the nearest hundredth. Record the fraction and decimal values in the table to the right. 4. Use the grid below to create a scatter plot of the x-values from the table and then draw a smooth curve connecting the points on y your scatter plot. x y = 1/x Decimal Value 1 1/ / / / / / / / / / x

99 AIIF Page 94 SJ Page What do you notice about the values of y, which represent the lengths of the strips, as the values of x increase? The values of y decrease as the values of x increase. 6. As the values of x get larger and larger, what value does y seem to approach? As the values of x get larger, the values of y get closer and closer to Using your graphing calculator, determine the values of y for each x value in the table below. Write the y values in the right column of the table. x y = 1/x 1/ 1/5 5 1/ / / / / / ,000 1/ ,000 1/ ,000, From your results from Exercise 7, as x gets closer to 0 what value does y get closer to? As x gets closer to 0, the values of y get closer to positive infinity. 9. Can we find the value of y for x = 0? Explain. We cannot find the value of y for x = 0 because we cannot divide by 0, the result is undefined. 10. Investigate variations of the inverse function by using your graphing calculator. a. Graph y = /x. Describe the differences between this graph and the graph y = 1/x. Answers will vary. A sample response might be: The graphs are similar. The values of y = /x are twice as large as the values for y = 1/x. b. Graph y = 3/x. Describe the differences between this graph and the graph y = 1/x. Answers will vary. A sample response might be: The graphs are similar. The values of y = 3/x are three times larger than the values for y = 1/x. c. Graph y = 1/x. Describe the differences between this graph and the graph y = 1/x. Answers will vary. A sample response might be: The graph of 1/x is in quadrants two and four compared to one and three for y = 1/x. The values are also opposites. d. Graph y = /x. Describe the differences between this graph and the graph y = 1/x. Answers will vary. A sample response might be: The graph of /x is in quadrants two and four compared to one and three for y = 1/x. The values of /x are opposite the values of 1/x and twice as large.

100 Non Linear Functions Lesson 5: Inverse Variation Activity 1 Grid Template y AIIF Page 95 SJ Page 41 1 unit Cut Here x

101 AIIF Page 96 Activity 1 Strip Cutouts SJ Page 43 Cut Here

102 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 97 Teacher Reference Activity In this activity, students will investigate real world applications of inverse variation. Tell the class that in the last activity, they looked at the equation y = 1/x and discovered that as x increased y decreased. The class should also realize from the last activity and Setting the Stage that as x increased y decreased proportionately and so the equation y = 1/x represents inverse variation. From this we can conclude that the converse is also true, as x decreases then y increases proportionally. Write the general inverse variation equation, y = k/x on the board or a blank transparency on the overhead projector. Write the words for the equation, "y varies inversely with x" and "y varies inversely proportional to x." Tell the class that k represents the constant of proportionality. Let the class know that inversely is the key word here. Another way of looking at the equation is xy = k, which says that the product is k, and is known as the constant of proportionality. Have the students investigate inverse variation and k by using the xy = k equation by doing Exercise 1. The students could work individually or in pairs. Have a volunteer model a similar problem, such as xy = 36, on the board or a blank transparency at the overhead. The class could discuss the answers for parts a through f together as a class. While the class is working, circulate to ask guiding questions and provide encouragement. Now, let's look at real world applications of inverse variation. Let the class know that k must be calculated from information given in the problem. Model the following problem with the class to find k: The number of hours, h, it takes for a block of ice to melt varies inversely with the temperature, t. If it takes hours for a square inch of ice to melt at 65 F, how long will it take for the ice to melt at 60 F? Have a volunteer come to the board or overhead projector and model solving this problem with the class assisting. Ask guiding question as needed. The first question is, "How do the variables in the problem match with the variables in the general inverse variation equation y = k/x?" The class should make the connection that since h will be increasing as the temperature decreases, h represents y and then t must represent x. Have the volunteer write the inverse variation equation, with class assistance if necessary, for the given variables in the problem. The student volunteer should write h = k/t. Ask the class, "What do we know from the problem and what don't we know?" The class should state that they know it takes hours for the ice to melt at 65 F, but they don't know how long it will take to melt at 60 F and they don't know the constant of proportionality k. Ask the class, "Do we solve for k first or do we solve for the time it takes the ice to melt at 60 F?" The class should realize that they can't solve for the time it takes the ice to melt at 60 F until they know the constant of proportionality k. Have the student volunteer solve for k with the class assisting as necessary. The result is that k = (65) = 130. After the student volunteer has solved for k, have her/him answer the original question in the problem by finding the time it will take the ice to melt at 60 F. Namely, h = 130/60.17 hours. Have a second and third volunteer model solving a problem while the class parallels with a similar, but different problem:

103 AIIF Page 98 Volunteer: Class: Volunteer: Class: y varies inversely with x. y = 7 when x = 6. Use the equation y = k/x. What is the value of y when x = 1? The volunteer should get that k = 4. Similarly we find that y = when x = 1. y varies inversely with x. y = 1 when x =. What is the value of y when x = 6? The class should get that k = 4. The class should find that y = 4 when x = 6. y varies inversely with the square of x. y = 18 when x =. Use the equation y = k/x. What is the value of y when x = 3? The volunteer should get that k = 7 and that y = 8 when x = 3. y varies inversely with the square of x. y = 6 when x = 4. What is the value of y when x = 8? The class should get that k = 96 and that y = 1.5 when x = 8. Discuss with the class the difference between the last two problems they solved. The key idea here is the x was squared in the last problem. Let the class know that y could also vary inversely with the cube of x, x 3, or y could vary inversely with the fourth power of x, x 4. Have students work in pairs on Exercises through 8. For Exercise 8, you might want to lead a short discussion and ask the class, "What sort of information in the graph can be used to write the inverse variation equation?" The class should realize that they have several ordered pairs in the graph which could be used to find the constant of proportionality and then use the value of k to write the inverse variation equation. Have student pairs check their results with other student pairs. Bring the class together and have volunteers share their results for any problems that the class had trouble with.

104 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 99 Activity SJ Page 45 In this activity, you will solve inverse variation problems and real world inverse variation problems. The general form for an inverse variation equation is y = k/x, where k is called the constant of proportionality. Another way of writing this equation is xy = k. You will investigate inverse variation and k by using the xy = k equation. 1. Find five different sets of values (ordered pairs) that make xy = 4 true. Answers will vary. A sample response might be: Five different sets of ordered pairs are (1, 4), (, 1), (4, 6), (, 1), and ( 8, 3) a. As the values of x increase what do you notice about the values of y? The answers may vary. A sample response might be: As the values of x increased the values of y had to decrease. b. Why would the values of y have to decrease as x increases to keep the equation true? The answers may vary. A sample response might be: If both the values of y and x increased then we wouldn't be able to keep the constant value of 4. c. If the x value doubles what happens to the y value? The answers may vary. A sample response might be: If the value of x doubled then the value of y would be half as much as it was previously. d. If the x value triples what happens to the y value? The answers may vary. A sample response might be: If the value of x triples then the value of y would have to be one third as much as it was previously. e. What happens to the relationship between x and y if we change the constant to a different number such as 36? The answers may vary. A sample response might be: Both x and y must adjust so that the product is now 36. f. Why do you think equations in the form of xy =k, where k is constant, are called inverse variation equations? The answers may vary. A sample response might be: Because as one variable increases the other must do the opposite or the "inverse" of increasing which would be decreasing. For Exercises and 3, use the given information to solve for the constant of proportionality k and then for the unknown value of y.. If y varies inversely with x and y = 34 when x = 1/68, what is the value of y when x =? The value of k is 0.5. Therefore, y = 0.5 when x =. 3. If y varies inversely with the cube of x and y = 10 when x = 4, what is the value of y when x =? The value of k is 640. Therefore, y = 80 when x =.

105 AIIF Page 100 SJ Page 46 For Exercises 4 through 7, use the information given in the problem to find the constant of proportionality k and answer the question. 4. The number of hours, h, it takes to mow a lawn varies inversely with the number of people mowing the lawn at the same time. a. If it takes 3 hours for 3 people to mow the lawn, how long will it take 5 people to mow the same lawn? The constant of proportionality k = 3(3) = 9. It will take 5 people 1.8 hours to mow the lawn, h = 9/5. b. Write an inverse variation equation for the problem. The inverse variation equation is h = 9/p, where p represents the number of people mowing the lawn. 5. Boyle's law states that in a perfect gas where mass and temperature are kept constant, the volume, V, of the gas will vary inversely with the pressure, P. A volume of gas, 550 centimeters cubed, is under a pressure of 1.78 atmospheres. a. If the pressure is increased to.5 atmospheres, what is the volume of the gas? The constant of proportionality k = 550(1.78) = 979. The volume of the gas is centimeters cubed, V = 979/.5. b. Write an inverse variation equation for Boyle's law. The inverse variation equation for Boyle's law is V = 979/P. 6. In hydraulics, the fluid pressure, P in pounds per square inch, is related directly with the force, f in pounds, and inversely with the area, A in square inches. The formula is P = f. Assume the force is A kept constant. a. If the fluid pressure is 5 pounds per square inch when the area is 0 square inches, what is the fluid pressure when the area is 40 square inches? The constant force is f = 5(0) = 100. The fluid pressure is.5 pounds per square inch when the area is 40 square inches, P = 100/40. b. Write an inverse variation equation for the fluid pressure. The inverse variation equation for the fluid pressure is P = 100/A.

106 Non Linear Functions Lesson 5: Inverse Variation 7. The weight of a body varies inversely as the square of its distance from the center of the Earth. AIIF Page 101 SJ Page 47 a. If the radius of the Earth is 4000 miles, how much would a 00- pound man weigh 1000 miles above the surface of the earth? The constant of proportionality k = 00(4000) = 3,00,000,000. The man weighing 00 pounds on Earth would weigh 18 pounds 1,000 miles above the surface of the Earth. b. Write an inverse variation equation for the weight of a body. The inverse variation equation for the weight of a body is W = /d. 8. Use the graph to the right, to write an inverse variation equation. The inverse variation equation is y = 500/x. y x

107 AIIF Page 10 SJ Page 48 Practice Exercises For Exercises 1 through 3, use the given information to solve for the constant of proportionality k and then for the unknown value. Write an inverse variation equation for each exercise. 1. If s varies inversely with t and s = 30 when t = 30, what is the value of s when t = 10? The value of k is 900. Therefore, y = 90 when x = 10. The inverse variation equation is s = 900/t.. If y varies inversely with the fourth power of x and y = when x = 3, what is the value of y when x = 0.1? The value of k is 16. Therefore, y = 1,60,000 when x = 0.1. The inverse variation equation is y = 16/x If j varies inversely with the square of l and j = 16 when l = 4, what is the value of j when l = 8? The value of k is 56. Therefore, j = 4 when l = 8. The inverse variation equation is j = 56/l. 4. The current, I in amps, produced by a battery varies inversely to the resistance, R in ohms, of the circuit to which the battery is connected. a. If the current is 0.5 amps when the resistance is 10,000 ohms, what will the current be if the resistance is reduced to 500 ohms? The constant of proportionality is k = 10,000(0.5) = 500. The current when the resistance is reduced to 500 ohms is 1 amp, I = 500/500. b. Write an inverse variation equation for the current of the battery. The inverse variation equation for the current of the battery is I = 500/R. 5. The intensity, I, of light observed from a source of constant luminosity varies inversely as the square of the distance, d, from the object. a. If the intensity of a light is lumens when the light source is 1.1 meters away, what is the intensity of the light if the source is 3 meters away? Round all answers to four decimal places. The constant of proportionality is k = (1.1) The intensity of the light when the source is 3 meters away is 0.00 lumens, I = /3. b. Write an inverse variation equation for the intensity of light, I, a distance d from the source. The inverse variation equation for the intensity of the light is I = /d.

108 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 103 SJ Page Lengths of radio waves vary inversely with radio wave's frequency. a. Radio station WJHU broadcasts their FM signal with a frequency of 88.1 MHz and has a wavelength of approximately 3.4 meters. Boston's famous WRKO AM radio station broadcasts their signals with a frequency of MHz. What is the wavelength of WRKO's broadcasts? NOTE: Round your k value to the nearest whole number and the wavelength to the nearest tenth of a meter. The constant of proportionality is k = 3.4(88.1) 300. The wavelength of WRKO's broadcasts are 441. meters, w = 300/ b. Write an inverse variation equation for the wavelength of radio waves. The inverse variation equation for the length of radio waves is w = 300/F.

109 AIIF Page 104 SJ Page 50 Outcome Sentences Inverse variation is I know when a problem is about inverse variation because For inverse variation, y as x The opposite of inverse variation is I still need help with

110 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 105 Teacher Reference Lesson 5 Quiz Answers 1. The value of k is 60. Therefore, a = 600 when b = 0.1. The inverse variation equation is a = 60/b. 15. The value of k is 15. Therefore, y = 1.5 when x = 100. The inverse variation equation is y =. x 3. It will take 7 workers 3 hours to unload the same cargo jet.

111 AIIF Page 106 Lesson 5 Quiz Name: For Problems 1 and, use the given information to solve for the constant of proportionality k and then for the unknown value. Write an inverse variation equation for each problem. 1. If a varies inversely with b and a = 15 when b = 4, what is the value of a when b = 0.1?. If y varies inversely with the square root of x and y = 3 when x = 5, what is the value of y when x = 100? 3. The amount of time it takes to unload a cargo jet varies inversely with the number of workers unloading the jet. If 3 workers take 7 hours to unload the cargo jet, how long will it take 7 workers to unload the same cargo jet?

112 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 107 Lesson 6: Exponential Functions Objectives Students will be able to determine the difference between power functions and exponential functions Students will be able to identity exponential functions involving growth and decay Students will be able to write functions involving exponential applications Students will be able to graph exponential functions Students will be able to model exponential functions using exponential regression Students will be able to solve exponential functions involving applications Essential Questions How do exponential functions behave in real world applications? Tools Student Journal Dry-erase boards, markers, erasers Graphing calculator and view screen Poster paper Construction paper Warm Up Problems of the Day Number of Days 3 days Vocabulary Base Growth Exponent Decay Exponential function Exponential regression

113 AIIF Page 108 Teacher Reference Setting the Stage Use your favorite grouping strategy to place the class in groups of 4. Place the Setting the Stage transparency on the overhead projector. Tell the class that the diagram under the first table could represent something such as arranging oranges in a square while the diagram under the second table could represent a population tree where each generation doubles. Tell the class to discuss and answer the questions from the transparency. Give each group a piece of poster paper. Tell the class to answer their questions on the poster paper. Give the class 5 to 7 minutes to answer the questions and place their results onto the poster paper. After the time has expired, have each group share their results and findings with the class. Some answers to the questions will vary. Here are some sample responses. 1. The function rule for the left table is y = x while the function rule for the right table is y = x.. The similarities are that each rule has a base, an exponent, the number, and the variable x. 3. The first function rule would be classified as a quadratic and the second would be classified as an exponential. Students may not give this response but may correctly guess it since the lesson is about exponential functions. 4. The conclusion about the function rule for the second table is that the values of y double because the base is. Discuss with the class the difference between the various group rules and how they would classify the two rules (functions) they wrote. Questions such as, "Are the functions from the same classification?" and "How would you classify each function rule you wrote?" The class should realize that the first function rule can be classified as a power function as well as a quadratic function, while the second function rule is classified as an exponential function. The students may not know the classification of the second function unless they make a connection that it might have something to do with the name of the lesson.

114 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 109 Setting the Stage Transparency x y x y Write a function rule for each table above.. Discuss, in your group, the similarities and the differences between each function rule. 3. How would you classify each function rule (linear, quadratic, power, etc)? 4. What are your conclusions about the function rule for the second table?

115 AIIF Page 110 Teacher Reference Pre-Reading Complete the following pre-reading with the students for the first part of the activity on growth. Write the word Bacteria on the center of the board or overhead. Circle the word and then ask the class to list other words or ideas that relate to the word. As the students give words, place the words around the outside of the circle. You are creating a word web and students should be familiar with this from English. Spend about five minutes discussing the word. Write the words "respiratory", bacteria, and lungs on the over head and ask students why the words respiratory, lungs, and bacteria may be in the same paragraph. Sample responses might be: The lungs are part of the respiratory system; bacteria can get into the lungs and cause serious problems to the respiratory system. Explain to students that they will be modeling the effect of bacteria on the respiratory system (lungs) and that they will be modeling this effect with mathematics. Remind students of the three methods of representing relationships: numeric, graphic, algebraic. Have volunteers describe each method. Activity 1 For the first part of this activity, students will model an exponential growth function. You could let the class know that different bacteria grow at different rates and for this activity, our rates are for modeling purposes only and may not reflect actual growth rates. There are 6 steps involved with the "experiment": Experiment Step 1: Working with a partner, have the students cut out 64 red squares of construction paper. Another color may be used. The squares should be 1 square inch or 1 square centimeter in size. Experiment Step : Have the class cut out the lungs template at the end of the activity. Experiment Step 3: Have students place one red square on the lungs so that it is contained within the lungs. This represents the initial amount of bacteria, a single cell. Tell the class that each hour the bacteria doubles. Experiment Step 4: Walk around to each pair to see their progress. You may want to ask the pairs, How long do you think it will take until all of your squares have been placed down on your lungs? Experiment Step 5: Check to make sure the pairs are recording the information correctly. Each pair should have similar results. Experiment Step 6: Ask the students if they see any similarities with their results and either one of the tables from the Setting the Stage. The students should realize that their table matches the second table from the Setting the Stage transparency. Have the pairs answer Exercises 1 through 5 and ask for students to share their results with the class. Ask the class, "How important is it to obtain medical assistance if problems persist in your respiratory system?" The class should realize that since the bacteria are doubling every hour that if left unchecked could cause major health issues. After the class completes the bacteria portion of the activity, ask them, "If there is bacteria growing and duplicating in a respiratory system, how would one remove them before the person gets real sick?" This is a good time to do the pre reading with the class. Write the word Antibiotics on the center of the board or overhead. Circle the word and then ask the class to list other words or ideas that relate to the word. As the students give words, place them around the outside of the circle forming a word web. Spend about five minutes discussing the word.

116 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 111 Write the word kidney on the overhead and ask students why the words kidney and antibiotic may appear in the same paragraph. You may want to discuss the function of the kidneys with your students. You could share information such as, "The kidneys are used to filter waste products and extra water from the blood. The kidneys filter about two quarts of waste per day." For antibiotic you could say, "Antibiotics kill bacteria to help prevent people from becoming sicker or to eliminate the disease completely." Explain to students that in the next portion of the activity they will model the effect of the kidney on antibiotics in the bloodstream with mathematics. For the second portion of this activity, students will model an exponential decay function. There are 4 steps involved with the "experiment": Experiment Step 1: Have the students work with a partner. Students should cut out 40 red and 0 blue 1 inch squares of construction paper. (Note: Students can use the red squares from the first portion of the activity and another color.) The students will need to remember which color represents the blood and which color represent the antibiotics. Experiment Step : Have the students place 0 red squares and 0 blue squares into a container (paper bag or box.) This represents a bloodstream that is half blood and half antibiotics. Although in real life the bloodstream would not contain 50% antibiotics, this will produce a model quickly that represents the way antibiotics leave the bloodstream. Walk around to each pair to see their progress. You may want to ask the pairs, In reality a person could not have 50% of the blood stream filled with antibiotics, but why might we model 50%? Experiment Step 3: Shake the container and randomly remove 10 squares. Replace them with 10 red squares. Check to make sure the pairs are recording the information correctly. Each pair should have similar results. Experiment Step 4: The students should repeat Step 3 ten times. The amount of antibiotics in the blood stream should be decreasing. Some groups might have zero antibiotics left in the blood stream. Before the students complete the exercises you may want to remind them that we use exponents to show 4 repeated multiplication. For example, 3 means (3)(3)(3)(3). Ask the class, "What does 3 x mean?" Have the students work in pairs on Exercises 6 through 9. Have volunteers share their results with the class.

117 AIIF Page 11 SJ Page 51 Activity 1 Bacterial Growth Respiratory System Model Respiratory sicknesses (infections), such as bronchitis and pneumonia, are caused by bacteria. Once bacteria gets in our lungs, they can duplicate (reproduce) at a certain rate. The following experiment will model the amount of bacteria present over time. In this experimental model, we will use small construction paper squares of one color to represent the bacteria. Experiment Step 1: Cut out 64 red construction paper squares. Each square should be the same size and shape. The best size is 1 inch by 1 inch or 1 centimeter by 1 centimeter. Use a ruler to draw the squares before cutting. Experiment Step : Cut out the lungs template at the end of the activity. Experiment Step 3: Place one red square on the lung template (any where inside the lung area.) This represents the initial amount of bacteria, a single cell. Note: Bacteria are actually very small in size. A single cell of bacteria is about 1/10,000 th of a centimeter. Experiment Step 4: Every minute, add enough red squares to double the amount you had previously. This represents the bacteria duplicating (reproducing itself) every hour. While you are waiting for each minute to end, count out the necessary squares that you will be adding for the next minute. Also, record the time and amount of bacteria present in the lungs in the table provided below. Experiment Step 5: Repeat Step 4 until all 64 squares have been placed "in" your lungs. Experiment Step 6: You should realize that your table matches the table from the Setting the Stage transparency. Table 1: Bacterial Growth Experiment Hour Bacteria Count

118 Non-Linear Functions Lesson 6: Exponential Functions 1. Create a scatter plot of the hours compared to the number of bacteria in the lungs. What type of pattern occurred in the scatter plot graph? Answers will vary. In general, the students should describe the points following an increasing pattern from left to right. It should be a non linear pattern. The students may describe this as a curve.. What is the rate that the bacteria are growing? The students should be able to determine that the bacteria's rate is doubling each hour. 3. Graph a scatter plot of your data on a graphing calculator. Set the window range to an x minimum of, x maximum of 7, x scale of 1, y minimum of, y maximum of 100, and y scale of 10. Is the scatter plot linear? If not describe the shape of the graph. Bacteria Count Bacteria Growth Experiment AIIF Page 113 SJ Page Hour Answers may vary. A sample answer might be: "No, the scatter plot is not linear. The scatter plot looks like half a parabola. Walk around to help pairs as needed when students are completing the table and graphing the data. 4. How many bacteria do you expect to be in the lungs after a 4 hour period? How might you calculate this value? Sample response: To calculate the value I would use an exponential function with as the base and 4 as the exponent. There should 4 or 16,777,16 bacteria in the lungs after a 4-hour period. 5. Approximately how many hours will it take until there are 1 trillion (1,000,000,000,000 or 1 x 10 1 ) bacteria in the lungs? NOTE: The graphing calculator may display 1 trillion as 1.0 E1. It will take approximately 40 hours until 1 trillion bacteria are in the lungs.

119 AIIF Page 114 SJ Page 53 Antibiotic Decay in the Blood Stream Experimental Model To help cure illnesses antibiotics and/or medicines taken into the body are circulated throughout the body by the bloodstream. The kidneys take the drugs out of the blood. We saw, from the first part of the activity, how bacteria can duplicate and create enormous amounts of themselves in a relative short period of time. Bacteria left unchecked can cause major health problems. Sometimes the only way to become healthy again is by the use of antibiotics. The following experiment will model the amount of antibiotics left in the bloodstream over time. In this experimental model, we will use small construction paper squares of one color to represent the blood and small construction paper squares of another color to represent the antibiotics. Experiment Step 1: Experiment Step : Experiment Step 3: Cut out 40 red construction paper squares and 0 blue construction paper squares. Each square should be the same size and shape. The best size is 1 inch by 1 inch. Use a ruler to draw the squares before cutting. Place 0 red squares and 0 blue squares in a container (bag or box). This represents a bloodstream that is half blood and half antibiotics. Although in real life the blood stream would not consist of 50% antibiotics, this will produce a model quickly that represents the way drugs leave the bloodstream. Shake the container and randomly remove 10 squares. Replace them with 10 red squares. Determine how many blood squares and antibiotic squares are now in the container. Place this information in Table 1 below. This step models the kidneys randomly cleaning one quarter of the blood each hour. Experiment Step 4: Repeat Step 3 ten times. Place the information for each cleaning cycle in Table, Antibiotics Decay Experiment, below. Table : Antibiotics Decay Experiment Hour Blood Count Antibiotic Count

120 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page Create a scatter plot of the hours compared to the number of antibiotics left in the bloodstream. What type of pattern occurred in the scatter plot graph? Answers will vary. In general, the students should describe the points following a decreasing pattern from left to right. It should be a non linear pattern. The students may describe this as a curve. 7. Create a transparency copy of your graph. Place all the transparencies from each group on the overhead at one time and line up the axes. What do you notice about the graph? Have each pair copy their graph onto a transparency and then align these on top of each other on the overhead so that they can see that most of the groups found a similar pattern. This should confirm to the students that it is a non linear pattern decreasing from left to right eventually reaching zero. Antibiotic Count Antibiotic Decay Experiment SJ Page Hour 8. If no new antibiotics are added, what would the graph do if we continued with the experiment? The students should be able to describe that the data or pattern will eventually reach zero and stay there. 9. Graph a scatter plot of your data on a graphing calculator. Set the window range to an x minimum of, a x maximum of 1, a y minimum of, and a y maximum of 4. Is the scatter plot linear? If not describe the shape of the graph. Walk around and help pairs as needed to complete the table and graph of the data. There may be some groups of students who have outliers that do not match the data. You may want to discuss with the class why that may happen. 10. Graph y = 0(0.75) x on the same graph as the scatter plot. Describe how the graph of y = 0(0.75) x fits the data from the scatter plot. Answers will vary. This function should model the data fairly well.

121 AIIF Page 116 SJ Page 55 Lungs Template Cut Here

122 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 117 Teacher Reference Activity In this activity, students will investigate the graphs and terminology of exponential functions. Students should be familiar with the terms base and exponent. Determine students understanding of these terms by placing the function rules, from the Setting the Stage, on the board or on a blank transparency on the overhead. The class should be able to tell you for the first function rule that x represents the base and is the exponent, y = x. For the second function rule, the class should be able to tell you that x represents the exponent and is the base, y = x. The class could also develop the standard form of an exponential function from the bacteria growth from the last activity. Hopefully, during your discussion of the differences between the function rules in the Setting the Stage you touched upon the topic that the exponent, for the function rule for the second table, was a variable not a constant. Tell the class that when the exponent contains a variable we have an exponential function. Have the class investigate the graphs of exponential functions in their groups. You may want to use the examples below to start. It is best if the students study one form, growth, and then move to the other form, decay. They did see an example of each type from the last activity. Sample exponential functions related to growth y = x and y = 5 x y = x x and y = 3( ) y = x x and y = + 3 y = x x and y = 3 y = x x and y = 3( ) + 1 Have the students form groups of three or four. Pose the following questions and have the students report their findings. Allow about minutes a question for discussion. Some guiding questions for their groups are: "What do you notice about the graphs as the base increases?" and "What do you notice about the graphs when the function is multiplied by a constant or a constant is added or subtracted?" Explain to the students that the graphs could have a vertical translation caused by the multiplication or addition/subtraction by a constant. Ask the students, "From our previous lesson about power functions, how did we obtain a horizontal translation of a function?" The students may remember that they had to add or subtract a value from x in order to obtain a horizontal translation. The same is true for exponential functions. These groups of exponential functions represent growth because y increases "exponentially" as x increases. Ask the groups, "How might we write an exponential function which decreases or decays?" Have students share their responses with the class. You might want to have a volunteer write the responses on the board. The students can use their graphing calculator to determine which methods produce a decreasing exponential function. Note: The functions should be exponential in nature, meaning the exponent should be a variable. Now have the class, in their groups, investigate the sample exponential functions below. These functions represent decay. Sample exponential functions related to decay

123 AIIF Page 118 y = x 1 and y = 3 y = 3 x 1 and y = y = 4 x 3 and y = 4 x x x Pose the following guiding questions for the groups to answer, "What do you notice about the graphs of the exponentials with negative exponents compared to the exponentials with fractions as bases?" and "Are exponentials with negative exponents and exponentials with fractions both examples of decay? Explain." Sample responses might be: The graphs of the exponentials with negative exponents are similar to the graphs of exponentials with fractions as bases. Yes, exponentials with negative exponents and exponentials with fraction bases are both examples of decay since as x increases, y decreases exponentially. Note: Since the graphing calculator will give a base with a fraction for exponential regression, the lessons will focus on exponentials with a fraction for the base instead of exponentials with negative exponents. Write the standard form of an exponential function, y = Ca, on the board. Note: y represents the function or dependent variable, C is a constant and also known as the initial amount when x = 0, a is the numeric base, b is a constant, and x is the variable exponent as the independent variable. Place the Activity transparency on the overhead projector. Ask for student volunteers to label the graphs as growth, decay, growth with initial amount, or decay with initial amount. The idea here is that students can look at an exponential function and should be able to determine if the function represents growth or decay and if there was an initial amount or not. Ask the class, "How many intercepts are there for exponential functions? What is the standard format for the y intercept of an exponential function?" The key goal here is that the class realizes there is only one intercept, the bx y intercept, and its standard format is (0, C). But in general, we use the standard form of y = Ca to define an exponential function. Use the sample examples below and model, or have volunteers model, how to determine the y intercept. Review how the students found the y intercept when they graphed equations of the form y = mx + b. Students should remember that for the y intercept the x coordinate has a value of zero. Also, have the volunteer state the type of graph the function represents. Sample Exponential Functions y = 4 x ; y intercept of (0, 1), type: growth x 1 y = 4 ; y intercept of (0, 4), type: decay with initial amount 3 3x y = 100(4 ) ; y intercept of (0, 100), type: growth with initial amount x+ 1 y = 1(5 ) ; y intercept of (0, 60), type: growth with initial amount Make sure the Activity transparency is displayed while to class works on the exercises as a reference. Have the class work in pairs on Exercises 1 through 6. Divide the board into 6 sections and have 6 students share their results with the class on the board. Ask the volunteers some guiding questions such as, "How did you bx

124 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 119 find the y intercept for your function?" and "How did you determine the type of graph and use it to draw your rough sketch of the function?" Some of the students might have answers such as, "I found the y intercept by setting x = 0 and then evaluating the function." and "I entered the function into my graphing calculator and viewed that the graph was increasing or decreasing so I knew it was either a growth or a decay type of graph."

125 AIIF Page 10 Activity Transparency A. y = Ca x, a > 1, C = 1 B. y = Ca x, a > 1, C > 1 C. y = Ca x, a > 1, C > 0 D. y = Ca x, 0 < a < 1, C > 0

126 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 11 Activity SJ Page 57 In this activity, you will determine the y intercept, determine the type of graph, and draw a rough sketch of exponential functions. For Exercises 1 through 4: a. Determine the coordinates of the y intercept b. Type of graph: growth or decay c. Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale appropriately. 1. y = 1 x y a. The coordinates of the y intercept are (0, 1). b. The graph is a growth type y = 8 x a. The coordinates of the y intercept are (0, 1). b. The graph is a decay type of graph x y 3. x 1 y = 9 5 a. The coordinates of the y intercept are (0, 9). b. The graph type is decay with initial amount x 4. y = 7(4 ) a. The coordinates of the y intercept are (0, 7). b. The graph type is growth with initial amount x y y x 100 x

127 AIIF Page 1 SJ Page 58 For Exercises 5 through 7, state the y intercept and the type of graph y 300 y x 100 x a. The coordinates of the y intercept are (0, 6). b. The graph type is exponential decay with initial amount. a. The coordinates of the y intercept are (0, 1). b. The graph type is exponential growth y 100 x a. The coordinates of the y intercept are (0, 3.5). b. The graph type is exponential decay with initial amount.

128 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 13 Teacher Reference Activity 3 In this activity, students will investigate real world applications of exponential functions. Explain to the class that in the last activity, they looked at the functions of the form y = Ca bx, where C was the initial value, a > 0. The values of b determined if the function was for growth (b > 0) or decay (b < 0). Either you or a volunteer model the following sample problem. The future value, S, of an investment that is compounded monthly; can be determined by the function S= P(1 + i) n, where P is the amount invested, i is the interest rate per month (rate/1) as a decimal value, and n represents the number of months the money has been invested. Determine the future value of a $15,000 investment, invested at the rate of 3%, for 60 months. From the information given, we see that P = $15,000, i = rate/1 = 0.005, and n = 60. Substituting in these 60 values and rounding to the nearest cent we get S = 15000(1.005) = $17,44.5. Have a second student volunteer model the following problem. From the bacterial growth problem from the first activity, we saw that the bacteria doubled every hour and the number of bacteria after t hours was given by the function y = t. If a different strand of bacteria t was present such that it quadrupled every hour, the function would be y = C(4 ), where C the initial amount of bacteria. Calculate the number of bacteria present after 4 hours if there are initially 10 bacteria present. From the information given, we see that C = 10, t = 4. Substituting in these values we get that the number of bacteria present after 4 hours is y = 10(4) X There is almost 3 quadrillion bacteria. Ask the class how this amount compares to their result in Exercise 4 from the first activity. Ask the class to calculate the ratio of the amount of these bacteria to the amount in Exercise 4 Activity 1. The class should get that there is 167,77,160 times more bacteria from this problem compared to Exercise 4 Activity 1. These are mind boggling numbers. Divide the class into teams of two. This might be different partnerships than before. Have the pairs work on Exercises 1 through 6. As the students finish their problems have pairs check their results with one or more other pairs. Bring the class together and have volunteers share their results on the board or overhead projector on any problems that the class had trouble. While the students are working, circulate to provide help, ask guiding questions, and provide encouragement. Remind the students that when using their calculators they should always double check their results. Note: For the Practice Exercises, you may want the class to work in pairs for Practice Exercise 4 through 8. You may want to make Exercise 7 optional or a bonus exercise.

129 AIIF Page 14 SJ Page 59 Activity 3 In this activity, you will solve real world exponential problems. 1. Your grandparents put $10,000 in an investment account, which collects interest four times a year, when you were born for your college education. The future value of your college education fund can be 4 determined by the function S = 10000(1.0375) t, where t represents the number of years for the investment. How much money will you have available when you start college? Assume you will be 18 years old when you start college. Draw a rough sketch of the investment; set axis scales accordingly. y x I will have $141,66.0 available when I start college. Note: y axis scale is 6,000:1. Viruses can produce many more offspring than bacteria per infection. Some viruses produce at an t exponential rate related to the function v= C(100) h, where v represents the number of viruses, C represents initial population of viruses, t represents amount of time in hours, and h is the number of hours to produce a new generation. How many viruses will be present after 4 hours if there initially were 5 viruses and the viruses produce a new generation every 4 hours? There will be 5,000,000,000,000 viruses after 4 hours. 3. It has been determined that a certain city has been growing exponentially over the last 0 years according to the function P= P0 (1 + r)t, where P represents the town's population, P 0 is the initial population, r is the rate at which the town's population is increasing, and t is the amount of time in years that the town has been increasing. If the town initially had 450 people 0 years ago and they now have 1,443 people, what was the rate of increase in population over the last 0 years? Round your answer to the nearest whole percent. The rate of increase for the population over the last 0 years was approximately 6%. 4. A local retail store has determined that its sales could grow exponentially based on the amount they spend on advertising each week by the function s= C(1.15) w, where s represents the number of sales per week, C represents their initial sales before advertising began, w represents the number of consecutive weeks they advertised. If the store averaged 15 sales per week before advertising began, how many sales can they expect to have, each week, after advertising for 4 consecutive weeks? Round your answer down to the nearest whole sale. The store can expect to have 18 sales each week after advertising for 4 weeks.

130 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 15 SJ Page The radio active decay of a material is given by the function ( t/t A = A ) 0e, where A 0 is the initial amount of the material, t is the amount of time in years, and T is the half life of the radio active material. Plutonium 40 has a half life of 6540 years. If a nuclear power plant started with 100 pounds of Plutonium 40, how much would be left after 0 years? How many ounces of plutonium decayed during the 0 years? Round your answers to the nearest hundredth pound and ounce. There would be pounds still left after 0 years. There was approximately 3.36 ounces of plutonium that decayed during the 0 years.

131 AIIF Page 16 Teacher Reference Activity 4 (optional) In this activity, students will learn and use exponential regression to determine the exponential function represented by a set of data. Review with students, if necessary, how to enter data in their graphing calculator. Since the students have used linear and quadratic regression models, they should be familiar with all aspects of regression except for maybe how to obtain the results for exponential regression. Have a volunteer model exponential regression by using the following data and the graphing calculator view screen. Time in minutes (t) Temperature C (T) After the class has entered the data, the class should press the following key sequence to obtain the exponential regression function STAT ~ Ê Í. The following screen shots coincide with all but the first key pressed. You could have the students complete a scatter plot of the data and have them store the exponential regression in the Y= editor by using the following key sequence: STAT ~ Ê ~ Í Í Í. The following screen shots show the majority of the above key sequence along with the graph and scatter plot. Have the class continue to work in their pairs for Exercises 1 through 3. Have students share their results using the graphing calculator view screen.

132 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 17 Activity 4 SJ Page 61 In this activity, you will use exponential regression to obtain an exponential function from real world data. 1. The following data table represents the daily costs of commuting (driving to work) versus the amount of commuters (people who drive to work) for a large metropolitan area. Cost (in $) Commuters 5, , ,000 68,000 35,000 13,000 8,000 5,600,500 a. What type of graph does the data model? The data models a decay type of graph. b. What is the exponential regression function? Round values to three decimal places. The exponential regression function is y = ( 0.891) x. c. How many commuters would you expect if they had to pay $75.00 each day in commuting expenses? Round your answer to the nearest commuter. There would be about 158 commuters.. The following data table represents the population of the United States from the years 1790 through 000, where year 0 = 1790, 1 = 180, etc. Year 0 (1790) 1 (180) (1850) 3 (1880) 4 (1910) 5 (1940) 6 (1970) 7 (000) Population (in millions) a. What type of graph does the data model? The data models a growth type of graph. b. What is the exponential regression function? Round values to four decimal places. The exponential regression function is y = ( ) x. c. Using this exponential equation, what might you predict will be the size of the U. S. population in the year 060? Round your answer to the nearest ten thousandths. Note: Remember our current units for population is in millions. Sample response: There would be about 1,365,730,000 people or 1, million people.

133 AIIF Page 18 SJ Page 6 3. The following table represents the early production of crude petroleum in the United States. Year 0 (1859) 10 (1869) 0 (1879) 30 (1889) 40 (1899) Oil Production (in barrels),000 4,15,000 19,914,146 35,163,513 57,084,48 a. What type of graph does the data model? The data models a growth type of graph. b. What is the exponential regression function? Round values to three decimal places. The exponential regression function is y = ( 1.54) x. c. U. S. oil production peaked in What could you predict was our country's peak output of oil in 1970? Round your answer to the nearest whole barrel. The peak output of oil production in the U. S. was approximately,819,000,000,000,000 or X10 barrels of oil. d. The actual U. S. oil production in 1970 was approximately 3,500,000,000 barrels. What can you say about your predicted value of production compared to the actual value of production? Answers will vary. A sample response might be: The values differ by quite a lot. e. What suggestion would you make on limiting the use of your exponential regression function? Answers will vary. A sample response might be: I think the function should be limited to within only a few years from the last date of data collection.

134 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 19 Practice Exercises SJ Page 63 For Exercises 1 and 3: a. Determine the coordinates of the y intercept. b. Type of graph: growth or decay. c. Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale appropriately. 1. y = 53 ( ). a. The coordinates of the y intercept are (0, 5). b. The graph type is growth. Note: y scale is 5:1 y x x 1 y = 9. 6 a. The coordinates of the y intercept are (0, 9). b. The graph type is decay with initial amount. Note: y scale is 100:1 y x = 3( 4 x ) y. a. The coordinates of the y intercept are (0, 3). b. The graph type is growth with initial amount. y x

135 AIIF Page 130 SJ Page On January 15 th, 009, the world's population was 6.75 billion people. It is predicted that it will take just 44 years for the world's population to double. What is the rate, per year, at which the world population is increasing? Round your answer to the nearest tenth of a percent. Note: Use P= P0 (1 + r)t. Refer back to Activity 3, Exercise 3. The world's population is increasing at an approximate rate of 1.6% per year. 5. A biologist is conducting an experiment testing a new antibiotic on a certain strain of bacteria cells. According to the biologist's calculation, the cells are dying (decaying) at a rate given by the 0.3t function L= ae, where L represents the amount of cells left after time t (in minutes) and a represents the initial amount of bacteria cells present before the antibiotic is applied. How many bacteria cells are present ten minutes after the antibiotic was applied if there initially were 10 million bacteria cells? Round your answer to the nearest whole cell. Use.178 for the value of e. There are approximately 1,075,84 cells left 10 minutes after the antibiotic was applied. 6. A person invests $15,000 into an interest bearing account. After 10 years the person's investment is now worth approximately $5,966. Determine the annual interest rate if the future value of an investment can be determined with the function S = P(1+ r/1) 1t, where S is the value of the investment after t years, P is the amount invested, and r is the annual interest rate. Round your answer to the nearest tenth of a percent. The annual interest rate is 5.5%. 7. The intensity of earthquakes is measured by using the Richter scale. We can determine how much more powerful one earthquake is compared to another earthquake, by the ratios of their intensities. The ratios of the intensities of two earthquakes can be determined by the function I = 10 d, where I is the ratio of intensities and d is the absolute value of the difference of the intensities of the earthquakes as measured by the Richter scale. It is estimated that the 004 Indian Ocean earthquake measured 9. on the Richter scale. In comparison, the earthquake that caused Mt. St. Helen's volcano to erupt on May 18 th 1980, measured 5.1 on the Richter scale. a. How much more powerful was the 004 Indian Ocean earthquake compared to the 1980 Mt. St. Helen's earthquake? Round your answer to the nearest whole number. The 004 Indian Ocean earthquake was 1,589 times more powerful than the 1980 Mt. St. Helen's earthquake. I = 10 (9. 5.1). b. What can you conclude about the difference in the intensities of two earthquakes? Answers will vary. A sample response might be: "The difference of the intensities of two earthquakes grows exponentially by a factor of 10.

136 Non-Linear Functions Lesson 6: Exponential Functions 8. For the following graph: AIIF Page 131 SJ Page 65 a. State the y intercept. The y intercept for the graph is (0, 10). b. State the type of graph. The graph type is decay with initial amount. y x

137 AIIF Page 13 SJ Page 66 Outcome Sentences Exponential growth is I know an exponential problem when Exponential decay is When graphing exponential functions The part about exponential functions I don't understand is

138 Non-Linear Functions Lesson 6: Exponential Functions AIIF Page 133 Teacher Reference Lesson 6 Quiz Answers 1. a. The coordinates of the y intercept are (0, 1). b. The graph type is growth with initial amount. c. Note: vertical scale is 10.. Time (years) Value of Investment ($) 0 5, , , , , $ 60,000 50,000 40,000 30,000 0,000 10, t (in hours) Size of virus colony ,953, ,88, ,0,703, t

139 AIIF Page 134 Lesson 6 Quiz Name: 1. y = 3(4 x+1 ). a. Determine the coordinates of the y intercept. b. Type of graph: growth, decay, growth with initial amount, or decay with initial amount. c. Draw a rough sketch of the exponential function on the grid provided. Note: set grid scale appropriately. Complete one of the two problems below. 4t r. The function S= P 1 + can be used to determine the 4 future value of an investment that is compounded quarterly (4 times per year). If $5,000 is invested at a rate of 4.5%, how much will the investment be after 5 years, 10 years, 15 years and 0 years? Create a table for the future value of the investment and create a connected scatter plot of the investment over the 0 year period. Time (years) Value of Investment ($) v= C 5 t, where v represents the amount of the virus after t hours and C represents the initial size of the virus colony. Create a table showing the size of the virus colony every 8 hours for a day period if there is initially a population of 5 virus cells. 3. The rate at which a particular virus duplicates is given by the function ( ) /8 t (in hours) 0 Size of virus colony

140 Non Linear Functions Lesson 7: Step Functions AIIF Page 135 Lesson 7: Step Functions Objectives Students will be able to determine the rise and run of a step function Students will be able to write equations involving step functions by using the floor function Students will be able to write equations involving step functions by using the ceiling function Students will be able to graph step functions using the int() function Students will be able to solve step functions involving real world applications Essential Questions How do step functions apply to real world applications? Tools Student Journal Setting the Stage transparency Dry erase boards, markers, erasers Graphing calculator and view screen Warm Up Problems of the Day Number of Days for Lesson Days Vocabulary Rise Greatest integer Ceiling function Run Floor function Binary number system Int () function Smallest integer Pitch/slope

141 AIIF Page 136 Teacher Reference Setting the Stage Place the class in pairs. Make sure each student pair has a dry erase board and two different colored markers. Place the Setting the Stage transparency on the overhead projector but cover the bottom portion containing the parts of a stair stringer. Tell the class that stairs in homes and on wooden porches usually are built with stringers cut from a single board. Ask the class to draw a set of similar stairs on the grid side of their dry erase boards. Have the class wait to draw the dotted line until all the parts of the stringer have been labeled. Uncover the bottom of the transparency containing the parts of the stair stringer. Have volunteers locate the place where the parts of the stair stringer belong on the transparency while the rest of the class locates the parts on the stairs they drew on their dry erase boards. Now have the class draw the pitch line on their stringer which connects the top of each tread as shown in the transparency. Lead a guided class discussion on the similarities and differences between the stringer and finding the slope of a line. Ask guiding questions such as, "What part or parts of the stair stringer are similar to the formula for the slope of a line?" and "What does the rise and run of the stair stringer tells us about stairs that will be built with the stringer?" The key concept here is the class understands that the rise and run of the stringer gives the slope of the stairs that will be built and eventually people will walk up and down on. Have the class calculate the slope of their stairs. Tell the class that local building codes for their city determines pitch (slope) of a set of stairs. Normally, the pitch value is 7/11, meaning 7 inch rise and an 11 inch run. Ask for students to share the slope of their stairs. Does the pitch of their stringer fall within the "building code" value? Note: Different cities and states could have different building code requirements that stairs must meet. Research the building codes in your city and state and tell the class what the building code for stairs are and that the stairs in the Setting the Stage transparency would have to be modified to meet local building codes.

142 Non Linear Functions Lesson 7: Step Functions AIIF Page 137 Setting the Stage Transparency Stairs Stringer Stringers in homes and on porches are usually cut from a single board. Parts of a Stair Stringer: Rise Run Stringer Board Pitch Line

143 AIIF Page 138 Teacher Reference Activity 1 Have the class continue to work in their pairs. Ask the class, "How do you think a step function looks?" and "How might the rise and run of a step function be utilized in graphing a step function?" Have the class draw (graph) their interpretation of a step function on their dry erase board. Have students show and explain their interpretation of what the graph of a step function might look like. In this activity, students will investigate step functions using their graphing calculator and how step functions apply to the application of salary. The instructions below are based on a TI-83 or 84 Plus graphing calculator. If you use a different type of graphing calculator, consult the owner's manual for graphing an equivalent step function. Tell the class that the graphing calculator has only one function that operates as a step function. This function is called the int() function and it represents the greatest integer. The int(x) is the greatest integer less than or equal to x. For example, if x = 3.14 then the int(3.14) = 3. Let the class know that mathematically the correct term to use is floor function when dealing with the greatest integer of a value. Tell the class it is called the floor function because the integer portion of the number represents the floor, or smallest value that the number can have. Another way to view it is that the floor is below us, so we want the integer below the give value. This could help students to remember when to round down. To obtain the int() function to the home screen or to the Y= editor, press the following key sequence: ~. The int( function is displayed. The screen shots below represent the key sequence. Have the class use their graphing calculator to find the values in the sample exercises. Have a student or students model the first problem in each problem set from the sample exercises using the graphing calculator view screen while the class parallels with the second problem in each problem set from the sample exercises. Note: We'll use the mathematical term floor in the sample exercises and when discussing the greatest integer function. Sample exercises floor of 3.5 and floor of 4.6 (results should be 3 and 4, respectively) floor of 6.8 and floor of 8.9 (results should be 7 and 9, respectively) floor of 9.6 and floor of 1.05 (results should be 5 and 3, respectively) Model, or have a student model, graphing y = int(x) on the graphing calculator view screen while the class parallels on their graphing calculator. Tell the students to set the graphing window parameters to standard. Have the students press the r key and trace the values of x and watch as the y values change for

144 Non Linear Functions Lesson 7: Step Functions AIIF Page 139 particular x values. Now, have a student model y = 3int(x) while the class parallels with y = int(x). Again, have the class use the r key to determine how multiplying the greatest integer function by a constant value affects the x and y values. Ask the students, "How were the values of x and y affected by the multiplication of a coefficient?" The students should be able to tell you that the x values were unaffected but the y values doubled or tripled when the floor function was multiplied by a coefficient of or 3, respectively. Have other students model the first problem in each sample exercises below while the class parallels with the second problem from each of the sample exercises. Circulate to provide additional help to students who are having difficulty by asking guiding questions or offering encouragement. After each example discuss how the x- and y values changed from the basic y = int(x) function. Sample exercises y = int(x)/3 and y = int(x)/ y = int(x) + 3 and y = int(x) + y = int(x) 3 and y = int(x) y = int(3x) and y = int(x) y = int(x/3) and y = int(x/) After the modeling has been completed, have the class get in groups of four and write a list of their findings from the sample exercises. Have the class use the term floor function or greatest integer function when describing their findings. Lead a discussion with the class on their findings and have groups share their list of findings. Have the students walk around to see how their list compares with other groups. A list of their findings might include: Dividing the floor function by a number divides the y values by the same amount (as compared to the original floor function of y = int(x)). Adding or subtracting a constant value to the floor function creates a horizontal translation by the number of units that is added or subtracted. Multiplying the x values by a number in the floor function decreases the x interval for the y values by a factor that x was multiplied by. For example, y = int(x) had an x interval of length 1 while y = int(x) has an x interval length of 1/. Dividing the x-values by a number in the floor function increases the x interval for the y values by a factor that x was divided by. For example, y = int(x) had an x interval of length 1 while y = int(x/) has an x interval length of. Show the class how to graph step functions from the graphing calculator. For example, to graph, y = floor(x) or, y = int(x), there would be a closed circle on the left endpoint of the interval, but an opened circle on the right endpoint. The first graph at the right is how the graph is displayed on the graphing calculator. The second graph is how it would actually look if graphing by hand. Model this closed and open circle for the endpoints for the intervals: 3 to to 1 1 to 0

145 AIIF Page to 1 1 to to 3 Each tick on the x axis represents units and each tick mark on the y axis represents units for the second graph above. Students can see how this works by setting the TblStart to 3 and ΔTbl to 0.1 and viewing the table of values for y = int(x) on the graphing calculator. Model for the students how to write the intervals for x which define the y values. The students could write these as inequalities like they did in the Solving One- Variable Equations unit for one variable inequalities. An example would be 3 x < or [ 3, ). Working with their partner, have the students complete Exercises 1 through 3. Students can check their work with another pair. While the students are working, circulate to provide support, clarifications, and encouragement. Bring the class together and have students share their results on the board or overhead projector on any problems that the class had trouble with.

146 Non Linear Functions Lesson 7: Step Functions AIIF Page 141 Activity 1 SJ Page 67 In this activity, you will investigate the graphs of step functions using your graphing calculator and the int( ) function which represents the greatest integer function. Mathematically, the greatest integer function is called the floor function. The int(x) is the greatest integer less than or equal to x. If x = 3.14 then, in function notation, f (3.14) = int(3.14) = 3. Likewise, mathematically f (3.14) = floor (3.14) = 3, or y = floor (3.14) = Evaluate the floor function using your graphing calculator. Write the function down as it was entered in the calculator. a. y = floor (3.001) The floor of is 3. I entered int(3.001) into the graphing calculator. b. y = 3 floor (15.06) Three times the floor of is 45. I entered 3int(15.06) into the graphing calculator. c. y = 6 floor ( ) Negative six times the floor of is 10. I entered 6int( ) into the graphing calculator. d. y = 5 floor ( 5.045) + 3 Five times the floor of plus 3 is 7. I entered 5int( 5.045)+3 into the graphing calculator. e. y = ( floor( 13.45) ) The square of the floor of is 196. I entered int( 13.45)^ into the graphing calculator.. Your younger sister wants to earn some money. She asks you if you have any chores she can do. Write a step (floor) function for each scenario below. a. You pay your sister $1.00 for each half hour of work. y = floor (x) b. You pay your sister $1.00 for each fifteen minutes of work. y = floor (4x) c. Your sister wants $.50 for each hour of work. y =.5 floor (x) d. Your sister wants $4.00 for each hour of work. y = 4 floor (x) e. Your sister wants $.50 for each half hour of work. y =.5 floor (x)

147 AIIF Page 14 SJ Page Create a table of values, which include intervals for x and values for y, and write a step function for each of the graphs below. a The step function is y = 3 floor(x). y 5 10 x x intervals [ 4, 3) or 4 x < 3 [ 3, ) or 3 x < [, 1) or 3 x < 1 [ 1, 0) or 1 x < 0 [0, 1) or 0 x < 1 [1, ) or 1 x < [, 3) or x < 3 [3, 4) or 3 x < 4 y- values b y x x intervals [ 18, 1) or 18 x < 1 [ 1, 6) or 1 x < 6 [ 6, 0) or 6 x < 0 [0, 6) or 0 x < 6 [6, 1) or 6 x < 1 [1, 18) or 1 x < 18 [18, 4) or 18 x < 4 y-values The step function is y = 4 floor(x / 6).

148 Non Linear Functions Lesson 7: Step Functions AIIF Page 143 Teacher Reference Activity In this activity, we will investigate another step function, the ceiling function. Ask the class, "What is the opposite of the floor?" The students should know that the ceiling is the opposite of the floor. Now ask the class, "If the floor of 3.14 is 3, what do you think the ceiling of 3.14 equals?" Have a student record the class responses on the board. If nobody came up with 4, tell the class that the ceiling is one more than the floor. The ceiling of x is defined to be the smallest integer not less than x. Another way to view it is that the ceiling is above us, so we want the integer above the given value. This could help students to remember when to round up. Now tell the class, "The graphing calculator only has the int() function which we can use as the floor function. If the ceiling is one more than the floor, for non integer values, could we use the int() function to behave like a ceiling function? If so, would there be restrictions as to when we could use it and when we could not use it?" The students should be able to state that to use the int() function as a ceiling function, all they would have to do is to add 1 to the int() function to get the correct results, but the int() function can only be applied to non integer values of x. In symbolic notation we would have y = int(x) + 1. Have the class use their graphing calculator to find the values in the sample exercises. Have a volunteer(s) model the first problem of each sample exercise using the view screen and overhead projector while the rest of the class does the second problem from each of the sample exercises. Note: We'll use the mathematical term ceiling in the exercises and when discussing the least integer function. Sample exercises ceiling of 1.51 and ceiling of 9.16 (results should be 13 and 10, respectively) ceiling of 7.98 and ceiling of 4.39 (results should be 7 and 4, respectively) ceiling of 39.6 and ceiling of 3.05 (results should be 7 and 6, respectively) ceiling of 8 and the ceiling of 6 (results should be 8 and 6, respectively) Tell the class to graph a ceiling function on the graphing calculator, other than just the ceiling of x, using the int() function is complicated and beyond the scope of this class. The students could graph the ceiling of x using int(x), but multiples or a shorter interval is beyond the scope of this lesson. Tell the students that just like everything else they do, mathematicians have symbols for both the floor and ceiling functions. Floor Ceiling

149 AIIF Page 144 Have students model writing the previous exercises using the ceiling symbolic notation while the class parallels with their same exercises. Sample exercises ceiling of 1.51 and ceiling of 9.16 (results should be y = 1.51 and 9.16, respectively) ceiling of 7.98 and ceiling of 4.39 (results should be y = 7.98 and 4.39, respectively) ceiling of 39.6 and ceiling of 3.05 (results should be y = 39.6 and y = 3.05, respectively) ceiling of 8 and the ceiling of 6 (results should be y = 8 6, respectively) Have the class work in pairs on Exercises 1 and. Have students check their results with one or more other pairs. Bring the class together and have pairs share their results on the board or overhead projector using the calculator view screen. While the students are working, circulate to provide assistance and to clarify questions. Bring the class together after the first two exercises have been completed. In the second part of this activity, students will investigate real world step functions. Remind the class that to expand a run by a certain amount, divide x by that amount inside our step function symbol. To increase the rise by a certain amount, multiply by that amount outside of the step function symbol. Remind the class about the rise and run that was discussed in the Setting the Stage activity. We ll use the same terminology for the example below. The terminology is used to assist the students in their understanding of the ceiling function and is not necessarily the standard terminology used when discussing the ceiling function. Let's take a look at an example involving the pay for a job. Have the class work in pairs. Tell the class to assume they have a part time job after school making $10 per hour and that they work 10 hours each week. Tell the class to assume you have to work a full hour to get any money. Ask the class to create a table of values that represent hours worked and pay in dollars for their part time job on the blank side of their dry erase board. Have a volunteer share their data table with the class on the board or on a blank transparency on the overhead projector. Ask the class, "What is the rise and run of your part time job?" The class should agree that the rise is $10 and the run is one hour. Now have the class write a step function equation for their job and remind them they have to work a full hour for payment for each hour. Ask the class, "Will the step function be a floor or ceiling? Explain." The class should agree that the function is a floor step function since they must work a full hour to get any pay. Ask for students to share the step function equation they wrote. The class should agree that the step function is y = 10 x. Have the student pairs graph their step function on the grid while a volunteer graphs their step function on a grid transparency. Tell the class to set a scale of 10 for the vertical axis, y axis. This means that each vertical tick mark equals $10.

150 Non Linear Functions Lesson 7: Step Functions AIIF Page 145 x (hours worked) y value (Pay in dollars) [0, 1) 0 [1, ) 10 [, 3) 0 [3, 4) 30 [4, 5) 40 [5, 6) 50 [6, 7) 60 y x Note: Each tick mark on the x axis represents 1 unit and each tick mark on the y axis represents 10 units Now ask the class, "What if we received pay for each 30 minute interval we worked. Would the run be the same? Would the run be expanded or contracted? Would the rise be the same? What would the new rise and run values be for the job?" Have students share their thoughts on the questions. Students should understand that the run and the rise will be different because we are earning the same amount per hour but we are getting half the pay every 30 minutes. Have the class write a new step function equation based on their new rise and run values and ask for students to share their new step function. The class should agree on the new step function, y = 5 x. Have the students graph their new step function and ask them what they should use for the x and y scales or what units should they label on the axes. Have a volunteer graph the step function equation on a grid transparency. 300 y x (hours worked) y value ($) [0, 0.5) 0 [0.5, 1) 5 [1, 1.5) 10 [1.5, ) 15 [,.5) 0 [.5, 3) 5 [3, 3.5) x Note: Each tick mark on the x axis represents 0.5 units and each tick mark on the y axis represents 5 units. Now have pairs group with another pair to form a group of four but remain in their pairs. Have one pair do the first example problem while the other pair does the second example problem. After the problems have been completed, tell groups to exchange problems and check each other s work. The pairs should write and graph a new step function based on the examples below. Examples Your pay is based on 1 minute work intervals Your pay is based on 15 minute work intervals Have pairs share their results with the class. The class should agree that the step function equation for the first example is y = 5x and that the equation for the second example is y =.5 4x.

151 AIIF Page 146 x (hours worked) y value ($) [0, 0.) 0 [0., 0.4) [0.4, 0.6) 4 [0.6, 0.8) 6 [0.8, 1) 8 [1, 1.) 10 [1., 1.4) 1 x (hours worked) y value ($) [0, 0.5) 0 [0.5, 0.5).5 [0.5, 0.75) 5 [0.75, 1) 7.5 [1, 1.5) 10 [1.5, 1.5) 1.5 [1.5, 1.75) 15 y 300 y x 100 x Note: For the first graph each tick mark on the x axis represents 0. units and each tick mark on the y axis represents units. For the second graph each tick mark on the x axis represents 0.5 units and each tick mark on the y axis represents.5 units. Working with their partner, have the class work on Exercises 3 through 6. Have pairs check their results with one or two other pairs. If students are having trouble, have other student partnerships share their results.

152 Non Linear Functions Lesson 7: Step Functions AIIF Page 147 Activity SJ Page 69 In this activity, you will investigate evaluating another form of step function, the ceiling function, using your graphing calculator and the int( ) function. Mathematically, the ceiling of x is called the least integer function. It represents the smallest integer not less than x. Remember from Activity 1, the int(x) function, mathematically the floor function, is the greatest integer less than or equal to x. If x = 3.14 then the int(3.14) = 3. However, the ceiling of 3.14, or 3,14, equals Evaluate the ceiling function using your graphing calculator. Write the function down as it was entered in the calculator. a The ceiling of is 36. I entered int(35.001) + 1 into the graphing calculator. b Negative four times the ceiling of 6.43 is 4. I entered 4(int( 6.43) + 1) into the graphing calculator. c Eight times the ceiling of is 8. I entered 8(int(0.045) + 1) into the graphing calculator. d Three point five times the ceiling of 7.89 minus two is 6.5. I entered 3.5(int( 7.89)+1) into the graphing calculator. e The cube of the ceiling of 4.8 is 15. I entered (int(4.8) + 1 )^ 3 into the graphing calculator.. Check the results from the ceiling functions below. If any of the results are incorrect, give the correct result and state what may have caused the incorrect results. a = 3 The correct results should be 4. The person may have used the floor function instead of the ceiling function. b = 8 The correct results should be 7. The person may have used the floor function instead of the ceiling function. c = 140 The correct results should be 140. The person may have multiplied by +4 instead of 4. d = 4 The correct results should be 8. The person may have subtracted instead of adding.

153 AIIF Page 148 SJ Page 70 In this part of the activity, you will investigate real world applications of step functions. In Activity 1, the run was expanded by dividing x by a certain amount inside the step function and contracted (shortened) by multiplying x by a certain amount inside the step function. Also, the run was increased by multiplying by a certain amount outside the step function and decreased by dividing by a certain amount outside the step function. For example, in Exercise of Activity 1, if your sister was paid every half hour, you had to multiply x inside the floor function by to decrease the run, y = floor (x). Also, when your sister wanted $.50 for each hour of work, you had to multiply the floor function by.5, y =.5 floor (x). 3. You have recently graduated from college and have taken a job with a company. Your starting salary is $30,000 per year. The company pays its employees once a month. a. Write a step function equation based on the information in the exercise. Answers may vary. A sample response is: "The step function equation is y= 500 x, where x represents the month worked. b. Create a table of values for an appropriate x interval. Note: To expand a run by a certain amount, we divide x by that amount inside our step function symbol. To increase the rise by a certain amount, we multiply by that amount outside of the step function symbol. x (months) y value ($) [0, 1) 0 [1, ) 500 [, 3) 5000 [3, 4) 7500 [11, 1) 7,500 [1, 13) 30,000 c. Graph your step function equation. y 5,000 0,000 15,000 10,000 5, Months x

154 Non Linear Functions Lesson 7: Step Functions AIIF Page 149 SJ Page From Exercise 1, the company has decided to pay its employees weekly. Note: Graph a portion of your function. a. Write a step function equation based on the information in the exercise. Note: There are 5 weeks in a year. Answers may vary. A sample response is: "The step function equation is y = x, where x represents the week worked. b. Create a table of values for an appropriate x interval. x (weeks) y value ($) [0, 1) 0 [1, ) [, 3) [3, 4) [51, 5) 9,43.08 [5, 53) 30,000 y c. Graph your step function equation. 15,000 d. Do you think there are any weeks where the pay could be different? Explain. There could be weeks where the pay is one cent more due to rounding to the nearest cent. Final pay must equal a whole year's salary. 10,000 5, Weeks x 5. After graduating from college with a degree in meteorology, you have taken up a position with NOAA, the National Oceanic and Atmospheric Administration. Your first assignment is to introduce a new tornado scale to replace the current Fujita Scale table shown below. Due to temperature changes over the past several decades, NOAA has decided to make a more consistent range of wind values for tornados. A gale force tornado will now start at 50 miles per hour (mph) and the new scale will have increments of 50 mph. The scale will still go from F0 through F6. Note: This problem deals with a hypothetical situation. The Fujita Scale Wind Speed F Scale Number Tornado Classification (MPH) 40 7 F0 Gale tornado F1 Moderate tornado F Significant tornado F3 Severe tornado F4 Devastating tornado F5 Incredible tornado F6 Inconceivable tornado

155 AIIF Page 150 SJ Page 7 a. Write a step function equation based on the information in the exercise. Answers may vary. A sample response is: "The step function equation is y = x / 50 1, where x represents the wind speed starting at 50 and going through 400, and y represents the scale from 0 through 6. y b. Create a table of values for an appropriate x interval. 6 5 x value (wind y (scale) speed in mph) [50, 100) 0 [100, 150) 1 [150, 00) [00, 50) 3 [50, 300) 4 [300, 350) 5 [350,400 ) 6 c. Graph your step function equation. Scale Wind Speed in MPH x d. What is the name you have given to your new tornado scale? Write the name of the new scale in the table above. The name of the new tornado scale will vary. Students may keep the current name or use their own name for the new tornado scale. 6. Computers store data using the binary number system, which has only two values, 0 and 1. Computers use voltages to record data as a 0 or a 1. Low voltages represent a 0 and high voltages represent a 1. Some computers use a RISC (Reduced Instruction Set Computer) microcontroller which operates in the range of voltages 0 to 18 volts. Assume that half the voltages represent a 0 (low voltages) and the other half of the voltages (high voltages) represent a 1. a. What is the interval of voltage values that would represent a binary value of 0 for a RISC based computer? The interval for low voltages would be [0, 9). 1 y b. What is the interval of voltage values that would represent a binary value of 1 for a RISC based computer? The interval for high voltage would be [9, 18). Binary Value c. Write and graph a step function representing the two binary values of 0 and 1 for the range of voltages. The step function would be y = x / 9. 9 Volts 18 x

156 Non Linear Functions Lesson 7: Step Functions AIIF Page 151 Practice Exercises SJ Page 73 For Exercises 1 through 4, evaluate the step function and state the type of step function, floor or ceiling. 1. y = 5.8 / The value of the floor step function is.5.. y = The value of the ceiling step function is y = The value of the floor step function is y = The value of the ceiling step function is 7. For Exercises 5 through 7, create a table of appropriate x- and y values based on the run and rise for the given step functions and graph the step function. Pick appropriate scales for your axes. 5. y = x /5 x y [-15, 10) 3 [ 10, 5) [ 5, 0) 1 [0, 5) 0 [5, 10) 1 [10, 15) The graph is a floor step function. 6. y = 5 x x y [ 1.0, 0.8) 5 [ 0.8, 0.6) 4 [ 0.6, 0.4) 3 [ 0.4, 0.) [ 0., 0) 1 [0, 0.) 0 [0., 0.4) 1 [0.4, 0.6) [0.6, 0.8) 3 [0.8, 1) 4 The graph is a floor step function.

157 AIIF Page 15 SJ Page y = 6 x /3 x y [ 1, 9) 4 [ 9, 6) 18 [ 6, 3) 1 [ 3, 0) 6 [0, 3) 0 [3, 6) 6 [6, 9) 1 [9, 1) 18 The graph is of a floor step function. For Exercises 8 and 9, create a table of appropriate x- and y-values for the given graph, write a step function, and state the type of step function, ceiling or floor. 8. x y [ 4, 3) 9 [ 3, ) 6 [, 1) 3 [ 1, 0) 0 [0, 1) 3 [1, ) 6 [, 3) 9 (3, 4) 1 The step function is y = 3 x 3 and it is a floor function.

158 Non Linear Functions Lesson 7: Step Functions AIIF Page 153 SJ Page x y [ 1, 9) 1 [ 9, 6) 8 [ 6, 3) 4 [ 3, 0) 0 [0, 3) 4 [3, 6) 8 [6, 9) 1 The step function is y = 4 x / and it is of type floor. For Exercises 10 and 11: a. Write a step function equation based on the information in the exercise. b. Create a table of values for an appropriate x interval. c. Graph your step function equation. 10. You have a part time job after school making $1.00 per hour. Your boss gives you partial pay for every 6 minutes that you work. a. Answers may vary. A sample response might be: "The step function is y = 1. x / 6. b. Possible table of values: x y ($) [0, 6) 0.00 [6, 1) 1.0 [1, 18).40 [18, 4) 3.60 [4, 30) 4.80 [30, 36) 6.00 [36, 4) 7.0 [4, 48) 8.40 [48, 54) 9.60 [54, 60) [60,66) 1.00

159 AIIF Page 154 SJ Page A company pays its employees a salary based on the number of years of employment with the company. New employees start with a salary of $5,000 a year. The company increases the employee s salary by $4, for each completed year of employment. a. Answers may vary. A sample response might be: "The step function is y = 4000 x b. Possible table of values: x (Years) y (Salary in dollars) [0, 1) 5,000 [1, ) 9,000 [, 3) 33,000 [3, 4) 37,000 [4, 5) 41,000 [5, 6) 44,000

160 Non Linear Functions Lesson 7: Step Functions AIIF Page 155 Outcome Sentences SJ Page 77 A step function is A floor step function is A ceiling step function is To increase the run of a step function To increase the rise of a step function I still need help with

161 AIIF Page 156 Teacher Reference Lesson 7 Quiz Answers y 1. Answers may vary. A sample table might be: 8 x y [ 4, 16) 1 [ 16, 8) 8 [ 8, 0) 4 [0, 8) 0 [8, 16) 4 [16, 4) 8 [4, 3) x The graph is a floor function.. Answers may vary. A sample table might be: x y [ 1.5, 1) 1.5 [ 1, 0.5) 1 [ 0.5, 0) 0.5 [0, 0.5) 0 [0.5, 1) 0.5 [1, 1.5) 1 [1.5, ) 1.5 The graph is a floor function. y x 3. The step function equation is y = 3 x / 5. Table of values is: x y [ 0, 15) 1 [ 15, 10) 9 [ 10, 5) 6 [ 5, 0) 3 [0, 5) 0 [5, 10) 3 [10, 15) 6 [15, 0) 9

162 Non Linear Functions Lesson 7: Step Functions AIIF Page 157 Lesson 7 Quiz Name: For Questions 1 and, create a table of appropriate x- and y-values based on the run and rise for the given step functions, graph the step function, and state the type of step function, ceiling or floor. Place correct labels and units on the graph. 1. y = 4 x/8. y = 1/ x 3. Create a table of appropriate x and y values for the given graph, write a step function, and state the type of step function, ceiling or floor. y x 5 10

163 AIIF Page 158 Lesson 8: Miscellaneous Non Linear Functions Objectives Students will be able to create a table of values for absolute value functions and circle equations Students will be able to determine vertical and horizontal translations of absolute value functions and circle equations Students will be able to write equations involving absolute value functions and circle equations Students will be able to graph absolute value functions and circle equations Students will be able to solve absolute value, piece wise, and circle equations which apply to real world applications Students will be able to write and graph piece wise functions involving real world applications Essential Questions How do absolute value functions, piece wise functions, and circle equations apply to real world applications? Tools Student journal Setting the Stage transparency Dry erase boards, markers, erasers Colored pencils Graphing calculator and view screen Activity transparency Activity 4 transparencies Warm Up Problems of the Day Number of Days for Lesson 3 Days (A suggestion is to complete Activity 1 and Practice Exercises 1 through 6 on the first day, Activity and Practice Exercise 7 though 10 on the second day, and then complete Activity 3 and 4 and remaining Practice Exercises and quiz on the third day.) Vocabulary Absolute value Transformation Translation Dilation Expansion Contraction Vertical line test Equation of a circle Piece-wise function

164 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 159 Teacher Reference Setting the Stage Divide the class into pairs. Make sure each pair has a dry erase board. Ask the students if they remember what absolute value means. See if the class can give some examples of absolute value. The key here is that students remember that absolute value represents the distance a number is from 0 on a number line and that distance is always a positive quantity. Display the Setting the Stage transparency. Ask for a volunteer to complete the table of ordered pairs, remind the class that they are finding the absolute value of x, and to create a scatter plot of the ordered pairs while the class does the same in their pairs. The class can assist the volunteer as needed. Ask the students what they notice about the shape of the scatter plot. Things they could say are, "The shape is like the letter V." Another should be, The left half represents a line with negative slope while the right half is a line with positive slope. They may also say, The left half is a mirror image of the right half (or vice versa) and the graph is symmetrical about the y-axis. A last possibility, The graph represents the graph of an even function. See if the class could write an equation for the table of ordered pairs. The class might be able to come up with y = x.

165 AIIF Page 160 Setting the Stage Transparency Absolute Value x y x y

166 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 161 Teacher Reference Activity 1 Have the class continue to work in pairs. In this activity, the class will investigate absolute value functions and translations for absolute functions. Ask the class, "We have investigated vertical and horizontal translations, from the origin, for other functions. How might we translate, vertically and horizontally, absolute value functions such as y = x?" Have student pairs get with another student pair to form a group of four. Have the groups discuss the question for two to three minutes and create a list of ideas and suggestions to the question. Have groups share their ideas with the class. Make sure all ideas have been presented. Have a student record the ideas and suggestions from the groups on the board or on a blank transparency on the overhead projector. Have the groups who give suggestions also give an example of their suggestions that they would model on a blank transparency grid on the overhead projector while the class models along on their dry erase board. Make sure the volunteers create a table of ordered pairs before drawing their graphs (basically they are creating a scatter plot of their suggested absolute value equation). Discuss the terms dilation, contraction and expansion with the class and how it relates to the graph of the absolute value function y = x. Students might be familiar with dilations from transformations in geometry. Use the examples y = x and y = x/. The first example will contract the graph of y = x while the second example will expand the graph of y = x. Discuss which ideas and suggestions translated the absolute value function y = x and which ones didn't. Have the class offer their interpretations of why some worked and why others may not have worked. Tell the class that the calculator uses the abs() function. This function can be found by pressing the following key sequence: ~ Í ( or À). The function abs( will appear. Have the class investigate the responses written on the board using their graphing calculator. Have student volunteers share the results of their investigations using the view screen graphing calculator. The following screen shots show how to obtain the abs( function on the TI 83 or 84 Plus graphing calculator. Have the class work in pairs on Exercises 1 through 17. Have pairs share their results with the rest of the class. Discuss general forms for translating an absolute value function vertically and horizontally as well as contracting and expanding absolute value functions.

167 AIIF Page 16 SJ Page 78 Activity 1 In the modeled exercises, you attempted to determine how to translate, vertically and horizontally, the absolute function y = x. In these exercises, you will continue your investigation of translations of y = x using the graphing calculator. For Exercises 1 through 8, you will investigate the transformations of the graphs of the absolute value function from the origin by adding, subtracting, multiplying, and dividing inside and outside the absolute value brackets. Using your graphing calculator, write an absolute value function for each absolute value situation, create a table of ordered pairs, and draw the graph on the provided grid. State the type and value of the transformation on the graph; vertical translation or horizontal translation compared to the graph of y = x. Also state if the graph has been dilated (contracted or expanded) compared to y = x. The first exercise has been completed for you. 1. Add 5 inside the absolute value brackets. The function is y = x + 5 ; the graph has been horizontally translated 5 units to the left. Grid scale is one horizontal and one vertical unit. y x y x. Add 5 outside the absolute value brackets. x y The function is y = x + 5; the graph has been vertically translated 5 units.

168 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page Subtract 3 inside the absolute value brackets. SJ Page 79 x y The function is y = x 3 ; the graph has been horizontally translated 3 units to the right from the origin. 4. Subtract 3 outside the absolute value brackets. x y The function is y = x 3; the graph has been vertically translated down by 3 units from the origin. 5. Multiply by inside the absolute value brackets. x y The functions is y = x ; y values have been contracted by a factor of.

169 AIIF Page 164 SJ Page Multiply by outside the absolute value brackets. x y The functions is y = x ; y values have been contracted by a factor of. 7. Multiply by inside the absolute value brackets. x y The functions is y = x ; y values have been contracted by a factor of. 8. Multiply by outside the absolute value brackets. x The functions is y = x ; y values have been contracted by a factor of. Y

170 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 165 SJ Page Was there a difference between Exercises 5 and 6? Explain your answer. There was no difference between Exercises 5 and 6 because we multiplied by a positive constant which did not affect the final results. 10. Was there a difference between Exercises 7 and 8? Explain your answer. There was a difference between Exercises 7 and 8 because we multiplied by a negative constant which reflected the graph about the x-axis when multiplied on the outside of the absolute value bracket. 11. What would you expect the results to be if we divided inside the absolute value brackets and outside the absolute value brackets by a positive constant? Explain your answer. Answers may vary. A sample response might be: "There would be no difference because we are dividing by a positive constant just like when we multiplied by a positive constant." 1. What would you expect the results to be if we divided inside the absolute value brackets and outside the absolute value brackets by a negative constant? Explain your answer. Answers may vary. A sample response might be: "There would be a difference because we would be dividing by a negative constant which would reflect the graph about the x-axis just like it did when we multiplied by a negative constant." 13. When dividing an absolute value function, inside or outside of the absolute value brackets, by a positive constant, what type of transformation would you expect on the graph: horizontal translation, vertical translation, dilation (contraction or expansion)? Explain your answer. Answers may vary. A sample response might be: "The type of transformation would be expansion by dividing by a positive constant because the y-values have been decreased by a factor of the positive constant we used to divide by." For Exercises 14 and 15, write a function for the given situation and draw its graph on the grid provided. 14. The function y = x has been horizontally translated left by units and vertically translated up by units from the origin. The function is y = x + +.

171 AIIF Page 166 SJ Page The function y = x has been expanded by a factor of and vertically translated down by 4 units from the origin. The function is y = x/ 4. For Exercises 16 and 17, write an absolute value function from the given graph The function for Exercise 16 is y = x 4 +5; the function for Exercise 17 is y = x

172 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 167 Teacher Reference Activity In this activity, students will investigate the equation of a circle and learn how to restrict its domain in order to make a function. Students will also be able to solve the circle equation for y in order to use the graphing calculator. Review with students the definition of a function and the Vertical Line Test. Display Activity transparency. Divide the class into groups of four using your favorite grouping strategy. Have the class discuss in their groups which shapes represent a function and which shapes do not represent a function. Tell the class to be prepared to explain their results. Give the groups 3 to 5 minutes for discussion. Have groups volunteer their results to the class. Each group should give their results for at least one shape. Ask the class, "How might we restrict the range for the non function graphs so that the graph is the graph of a function?" You may need to review the term range. Students can either discuss the question in their groups and then share results or you can lead a class discussion on the question. Have a volunteer record the class' responses on the board or on the activity transparency. Make sure the class agrees on how to restrict the range of each non function graph to make it the graph of a function. This concept is important so that the class can understand that we can make the graph of a circle a function if we restrict its range. Tell the class that the equation x + y = 100 has certain values that make it true. Using graphing calculators, have the students, in their groups, find integer values for x and y, that make the equation true. Ask groups to share one ordered pair of integer values that makes the equation true. Have a student list the group responses on the board or on a blank transparency on the overhead. Continue to ask groups to share their integer value ordered pairs until all ordered pairs from the table below have been shared. Have the groups plot the ordered pairs on a dry erase board and draw a connected graph as smooth as possible. Ask the class, "What do the connected ordered pairs form?" The students should realize that the connected ordered pairs form a circle. Ask the class, "What is the center of your circle?" and "How far from the center of the circle is each point on the circle?" These questions are to gauge the student's prior understanding of circles. This should include the center, the radius, and the fact a circle represents all the points that are equidistant from a point known as the center of the circle. x y Ask the students, "How could we rewrite our equation so we could enter it into the graphing calculator to graph?" and "What format must the equation be in so that we can enter it into the graphing calculator?" The goal is for students to understand that they must rewrite the equation into two separate equations y = 100 x and y = 100 x. You may need to assist students in rewriting the equations by reminding them of their equation solving skills from previous units. Have the class enter these equations into the graphing calculator to graph the circle. Let the class know that because the grid on the graphing calculator is not a square, the graph may not look circular. Another way to enter the two equations, at least for a TI 83 or 84 Plus graphing calculator, would be to enter the first equation in Y1 and then enter Y1 into Y. Now, ask the class, "We have seen many ways to translate functions from the origin. We know how to translate the graph of a parabola and power functions. In the previous activity we did transformations of absolute value functions. How might we write our equation of a circle to translate it horizontally and/or

173 AIIF Page 168 vertically from the origin?" You might want to give an example for a parabola (quadratic function) like y = x and y = ( x 4) to show a horizontal translation and y = x + 5 to show a vertical translation. Have the students discuss and investigate in their groups how to translate a circle from the origin. Have the class work with the equation x + y = 5. Tell the class they can either work on the equation with or without a graphing calculator. If they use a graphing calculator, let them know that they could just investigate "half" a circle by using the equation y = 5 x. Ask groups to share their results with the class. Have a student list the various methods shared by groups on the board or on the transparency used earlier. Make sure that information on the center of the circle after it has been translated was presented. If not, let the class know what the coordinates of a circle are after it has been translated. x h + y k = r. Tell the class that the coordinates (h, k) not only represents the horizontal and vertical translations, but also the center of the circle and that when the origin is the center of the circle we get the equation x + y = r. Write the general form of the equation of a circle on the board or blank transparency: ( ) ( ) Ask, "How many intercepts do you think a circle with its center at the origin has? Explain." The students could use their dry erase boards to assist them in answering the question. The key is that students understand that there are four intercepts and this concept can be used to draw a rough sketch of any equation of a circle centered at the origin. You could also ask, "How can the center of the circle and the circle's radius be used to help us plot points to graph the equation of a circle?" The center and radius can be used to plot four points that could be used as the cornerstone points to draw the graph of a circle. Have the class investigate graphing equations of a circle in their groups. Model, or have a student model, graphing the equation x + y = 16 on a transparency grid while the class parallels with x + y = 9 on their dry erase boards. A simple graphing strategy could be the following: Locate and plot the center of the circle at the origin Determine the value of the radius From the center (origin) of the circle, go up the number of units equal to the radius and plot a point. From the center (origin) of the circle, go down the number of units equal to the radius and plot a point. From the center (origin) of the circle, go left the number of units equal to the radius and plot a point. From the center (origin) of the circle, go right the number of units equal to the radius and plot a point. Draw the circle Have a volunteer(s) model graphing the first circle equation from the sample exercises below while the class follows along using the second equation from the sample exercises. The volunteer(s) should also find the coordinates of the center of the circle and four points associated with the center and radius r and solve their equation for y. Sample Exercises x + y = 4 and x + y = 9 x + y = 49 and x + y = 5 Remind the class about the general form of the equation of a circle, ( ) ( ) x h + y k = r, with center at (h, k). Ask the class, "How might we use the previous graphing technique for circles with the center at the origin to graph a circle with the center at some other location other than the origin?" The goal here is for students to

174 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 169 realize that after the center of the circle has been determined and a point plotted for it, they can use the same technique of going left and right, up and down by the value of the radius to sketch a rough graph of the circle. Have a volunteer(s) model graphing the first circle equation from the sample exercises below while the class follows along using the second problem from the sample exercises. The volunteer(s) should also find the coordinates of the center of the circle and four points associated with the center and radius r to graph the equation. Also, have the students solve the equation for y. Sample Exercises ( ) x 6 + y = 16and ( ) x y 3 + = 5 ( 7) x + y = 9and x ( y ) + 5 = 4 ( x+ 3) + ( y+ ) = 36and ( x ) ( y ) = 49 A simple graphing strategy could be the following: Locate and plot the center of the circle at the origin Determine the value of the radius From the center of the circle, go up the number of units equal to the radius and plot a point. From the center of the circle, go down the number of units equal to the radius and plot a point. From the center of the circle, go left the number of units equal to the radius and plot a point. From the center of the circle, go right the number of units equal to the radius and plot a point. Draw the circle Have the class work in pairs within their groups of four on Exercises 1 through 13. Have student pairs check their results with the other student pair in the group and possibly other groups. As you're walking around, note which groups have a good understanding and which groups need help. Have the groups that understand the content help the other groups by either going to those groups and assisting or sharing their results at the front of the class.

175 AIIF Page 170 Activity Transparency

176 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 171 Activity SJ Page 83 In this activity, you will investigate graphing the equations of a circle. For Exercises 1 through 6 (draw two circles per grid): a. State the radius and center of the circle. b. Draw a rough sketch of the circle equation. Use only four points to draw the rough sketch of the circle. State the coordinates of the four points. General Equation of a Circle with Center at (0, 0) and Radius r x + y = r General Equation of a Circle with Center at (h, k) and Radius r ( ) ( ) x h + y k = r 1. x + y = 81 a. The radius is 9 units and the center is (0, 0). b. The four points are ( 9, 0), (9, 0), (0, 9), and (0, 9).. x + y = 11 a. The radius is 11 units and the center is (0, 0). b. The four points are ( 11, 0), (11, 0), (0, 11), and (0, 11). x 6 + y = ( ) a. The radius is 6 units and the center is (6, 0). b. The four points are (0, 0), (1, 0), (6, 6), and (6, 6). 4. ( ) x + y+ 5 = 5 a. The radius is 5 units and the center is (0, 5). b. The four points are (5, 5), ( 5, 5), (0, 10), and (0, 0).

177 AIIF Page 17 SJ Page ( ) ( ) x+ 4 + y 3 = 49 a. The radius is 7 units and the center is ( 4, 3). b. The four points are ( 11, 3), (3, 3), ( 4, 10), and ( 4, 4). 6. ( ) ( ) x 6 + y+ 3 = 64 a. The radius is 8 units and the center is (6, 3). b. The four points are (, 3), (14, 3), (6, 5), and (6, 11). 7. Explain the technique you used to find the coordinates of the four points for circles that had a center at any location other than (0, 0). Answers will vary. A sample response might be: "After determining the coordinates for the center, I added and subtracted the value of the radius from the x coordinate of the center to get two points on a line parallel to the x axis. I then added and subtracted the value of the radius to the y coordinate of the center to get two more points on a line parallel to the y axis. For Exercise 8, pick any three equations from Exercises 1 through 6 and solve them for y. 8. Answers will vary. The equations in Exercises 1 through 6 solved for y are: y=± 81 x y=± 11 x y=± 36 ( x 6) y=± 5 x 5 y=± 49 ( x+ 4) + 3 y=± 64 ( x 6) 3

178 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page Using the general equation of a circle, ( ) ( ) SJ Page 85 x h + y k = r, with center at (h, k) and radius r, solve the equation for y to obtain a function equation for a circle. ( ) y=± r x h + k For Exercises 10 through 13, write the equation, in general form, for the graphed circle. State the center of the circle and the four points used to define its graph =, the center is The equation is ( ) ( ) The equation is x y 144 x y+ 7 = 49, the (0, 0) and the four points are ( 1, 0), (1, 0), center is ( 7, 7) and the four points are (0, -1), and (0, 1). ( 14, 7), (0, 7), ( 7, 0), and ( 7, 14).

179 AIIF Page 174 SJ Page 85 (cont.) The equation is ( x 1) + ( y 11) = 5, the The equation is ( ) ( ) x y 7 = 100, center is (1, 11) and the four points are (7, 11), the center is ( 7, 7) and the four points (17, 11), (1, 16), and (1, 6). are ( 7, 3), ( 7, 17), (3, 7), and ( 17, 7).

180 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 175 Teacher Reference Activity 3 In this activity, students will investigate piece wise functions, piece wise function with real world applications, and graphs of piece wise functions. Have the students continue to work as pairs in their groups of four. Ask the class, "How do you think a piece wise functions is constructed?" Students may say, "They are constructed in pieces." Technically, they are correct. Piece wise functions have different "definitions," formulas for different values of the domain. Write the following piece wise function on the board or on a grid 5 for x < 0 transparency on the overhead projector: y =. Have a volunteer graph the piece wise x+ 5 for x 0 function on the board or a grid transparency while the class graphs along on their dry erase boards and assists the volunteer as needed. This is an example of continuous piece wise function because the graph is not split or broken. Have other volunteers model the first of each sample exercises below while the class works on the second of each exercise below. Sample Exercises x+ for x < 0 y = x for x 0 x+ 5 for x < 0 and y = x 5 for x 0 3x+ 6 for x < 0 4x+ for x < 0 y = and y = x for x 0 x for x 0 3 for x < -5 8 for x < 4 y = x+ 1 for 5 x< 5 and y = x+ 5 for 4 x< 4 3x+ 9 for x 5 x+ 1 for x 4 Discuss the endpoints. Remind the students how they graphed endpoints for inequalities. Ask, "When do we have solid endpoints and when do we have hollow endpoints?" The class should remember their graphing techniques from inequalities, both one and two variable inequalities.

181 AIIF Page 176 The graphs of the sample exercises are displayed below.

182 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 177 These functions can also be graphed on the graphing calculator. It is important that the equations and the domain be kept inside separate parentheses when using the graphing calculator. Use the first sample exercise to demonstrate how to use the graphing calculator to graph piece wise functions. The instructions below are based on a TI-83 or 84 Plus graphing calculator. If you use a different calculator, consult the owner's manual for an equivalent function. The first equation is y = x + 1 for 0 < x. We would need to enter the equation inside parentheses, ( x + 1), and the domain inside parentheses as well, (0 < x). Also, remind the class that the inequality symbols are under the TEST menu. The key sequence is y then press the appropriate number key for the required inequality symbol. For the second equation, y = x 1 for x 0. We would need to enter the equation inside parentheses, (x 1), and the domain inside parentheses as well, (x 0). Set the graphing parameters to be: 1 for x minimum, 13 for x maximum, 1 for y minimum, 13 for y maximum, and x- and y-scale values of 1. The screen shots below show the equations in the Y= editor and the resulting graph. Note: The graphs should be dotted (not connected). Sometimes a dotted graph is easier to view and sometimes it makes it more difficult to view. Have the class work in pairs on Exercises 1 through 9. Tell the class they will need to write a piece wise equation for Exercises 5 through 7 from a graph and for Exercises 8 and 9 they will have to write and graph a piece wise equation from an application problem. Let them know that they have written many different equations for real world applications before. For these exercises, they will need to pay particular attention to the domain values for their piece wise equations, just like they did when they wrote inequality equations from real world application problems. Have volunteers share their results on the board or overhead projector with the class.

183 AIIF Page 178 SJ Page 86 Activity 3 In this activity, you will investigate piece wise functions and their graphs. You will also use the graphing calculator to graph piece wise equations. Follow your teacher's directions for graphing piece wise equations on the classroom graphing calculator. For Exercises 1 through 4, graph the piece wise equation on the grid provided. Also, graph the piece wise equation on your graphing calculator and write the format of the equation as it was entered into the graphing calculator. 1. 5x+ 3 for x < 0 y = x+ 7 for x 0 Answers may vary. A sample response might be: "I entered (5x + 3)(x < 0) in Y1 and ( x + 7)(x 0) in Y.". x for x < 0 y = 3x 6 for x 0 Answers may vary. A sample response might be: "I entered (x )(x < 0) in Y1 and (3x 6)(x 0) in Y." 3. x for x < y = 8 for x x Answers may vary. A sample response might be: "I entered (^x)(x < ) in Y1 and (8/x)(x ) in Y."

184 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page for x < 3 y = x for x< 4 for x Answers may vary. A sample response might be: "I entered (4)(x < ) in Y1, (x 3 )( x and x < ) in Y, and ( 4)( x) for Y3." SJ Page 87 For Exercises 5 and 6, write a piece wise equation for the given graph x + 1 for x < 0 The piece wise equation for the graph is y =. x for x > 0 9 for x < The piece wise equation for the graph is y =. x + 4 for x

185 AIIF Page 180 SJ Page 88 For Exercises 7 through 9, write a piece wise equation from the given information and then graph your piece wise equation. Set axes scales accordingly. 7. The a local electric company charges $ per kilowatt hour (KWH) for the first 00 KWH used and then $0.076 per kilowatt hour used beyond the initial 00 KWH. What does the value of y represent? The piece wise equation for the graph x for x 00 is y = (x 00) for x > 00 The value of y represents the total cost of electricity A cell phone company charges a $39.99 monthly fee that includes 500 anytime cell minutes. If you use more than 500 cell minutes, the cell phone company charges $0.40 for each additional minute. What does the value of y represent? The piece wise equation for the graph for 0 x < 500 is y =. 0.40(x - 500) for x 500 The value of y represents the total monthly cell phone bill The Reel Time movie theater charges $4.50 for children younger than 1 and for adults 65 and older. Everybody else must pay the full price of $10. The piece wise equation for the graph 4.50 for x < 1 is y = 10 for 1 x < for x

186 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 181 Teacher Reference Activity 4 Courting the Graphing Calculator (Optional) This activity is meant to be both fun and challenging. Have the students work as pairs in their group of four. Students will use their knowledge and understanding of the circle equation, written in y = format, and the graphing calculator to draw half an NBA size basketball court. Also piece wise equations will be used. The dimensions of an NBA basketball court are given in the student's journal. Display Activity 4 transparency. Point out that this view is from the end of the court to midcourt, the suggested view that they display on their graphing calculator. Point out the dimensions of the basketball court. Tell the students that they are only going to draw half the court on their graphing calculator. They can do the view from the end of the court to midcourt or from midcourt to end of the court. Tell them that they can use horizontal line equations (y =) to draw the boundary line, midcourt line, and the free throw line, and that the free throw line will require a piece wise equation and the inequalities will need to be written in compound form using AND (LOGIC menu under TEST). For example, the inequality -6 x 6 would have to be entered as 6 x AND x 6. Lead a discussion on the format of the equations needed to draw the half circles. Tell the class that some of the circles represent the top half or bottom of a circle. Ask the class, "What equation format would represent a lower half circle?" The class may remember that y = r x represented the upper half of a circle with center at the origin and the opposite, y = r x, would represent the lower half. Give the class recommendations for the graphing window parameters as shown in screen shots below. You might want to ask, "Why are we suggesting the x graphing window parameters range from 5 to 5, but the y graphing window parameters only have positive values?" The key concept is that writing the half circle equations is easier if we only need a vertical translation instead of both a vertical and horizontal translation in our equations. It also gives the students a line of symmetry to use. For the students to draw vertical lines they will need to use the DRAW menu (y <). Tell them to use the second option, Line(, and not the fourth option, Vertical, because Vertical will draw a vertical line the size of the calculator screen and they won't be able to control its length. The screen shots below show how to use the Line( option. Tell them that they can draw all of their vertical lines at one time. Also, it is important to tell the class to be in "graph" mode, meaning to press the s key before y <. To obtain the blinking + cursor to start drawing a line press the key sequence y < Á. Use the left, right, up, and down arrow keys to position the + cursor at the location where the line will start. Press the Í key to mark the location. The blinking + cursor will then change to a blinking rectangle cursor. Then use the up or down arrow key to move the rectangle cursor to the location for the end of the line and then press the Í key. Continue in such a manner to draw all the vertical lines at the same time. After the vertical lines have been drawn tell the class to press the s key to terminate DRAW mode. Have the class practice drawing multiple vertical lines before

187 AIIF Page 18 tackling the court challenge. Note: Students can use the Line( function from the home screen and enter endpoints instead of approximating them from the previous technique discussed. The format is Line(X1, Y1, X, Y) where (X1, Y1) and (X, Y) are the coordinates of the endpoints. The following screen shots show the equations in the Y= editor as well as the end result. Tell the students that because the rim is quite small and very close to the boundary line, they can use the Circle( option, option 9 in the DRAW menu, to draw a circle for the rim, but they should make it a little larger than it actually is and a little farther than the boundary line. A circle is drawn in the same manner as a line. Move the blinking + cursor to the location for the center of the circle, press the Í key to obtain the blinking rectangle cursor, and then use the arrow keys to position the cursor for the radius of the circle and press the Í key once more. Note: You can also draw a circle from the home screen using the format: Circle(X,Y,radius) where (X, Y) is the center of the circle and radius represents the radius of the circle.

188 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 183 Activity 4 Transparency three point line

189 AIIF Page 184 SJ Page 89 Activity 4: Courting the Graphing Calculator In this activity, you will use your knowledge and understanding of equations for horizontal lines, half circles, and piece wise functions to draw half a basketball court on your graphing calculator. Follow the instructions given by your teacher to draw vertical lines as needed on your calculator screen. The information and diagram below show requirements of a basketball court. The dimensions of an NBA basketball court are: Length of court: 94 feet Width 50 feet Diameter of rim: 18 inches Distance from backboard to free throw line: 19 feet Distance from backboard (boundary line) to rim: 6 inches Width of the key: 1 feet Three-point line/arc: From the center of the rim (basketball hoop) to the three-point line is.5 feet. From the center of the rim to the arc is 3.75 feet. three point line

190 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 185 Practice Exercises SJ Page 90 For each Exercises 1 and 4, create a table of ordered pairs based on the given absolute value function, graph the absolute value function, and state any horizontal or vertical translation from the origin and whether the graph has been expanded or contracted. Label the axis and units on the graph y 1. y = x+ 6 x y The graph is translated horizontally to the left by 6 units x. y = x 4 6 y x y The graph is translated horizontally to the right by 4 units and vertically down 6 units x 3. y = x/3 y x y The graph is expanded by a factor of x

191 AIIF Page 186 y SJ Page y = x x y x The graph is horizontally translated to the left by 3 units, vertically up by 1 unit, and contracted by a factor of. For Exercises 5 and 6, create a table of ordered pairs for the given graphs, write an absolute value equation, and state any horizontal or vertical translation from the origin and whether the graph has been expanded or contracted y 5 x x y The absolute value equation is y = x 4 3; horizontal translation of 4 units to the right and a vertical translation of 3 units. y 6. x y x

192 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 187 The absolute value equation is y = x/ + 1; vertical translation of 1 unit, expanded by a factor of. For Exercises 7 through 10 (draw two circles per grid): SJ Page 9 a. State the radius and center of the circle. b. Draw a rough sketch of the circle represented by the equation. Use only four points to draw the rough sketch of the circle. State the coordinates of the four points. c. Solve the equation for y. y 7. x + y = 4. a. The radius of the circle is. b. The four points for the rough sketch of the graph are (, 0), (, 0), (0, ), and (0, ); the center of the circle is (0, 0). c. y=± 4 x x+ 4 + y = ( ) a. The radius of the circle is 8. b. The four points for the rough sketch of the graph are ( 1, 0), (4, 0), ( 4, 8), and ( 4, 8); the center of the circle is ( 4, 0). c. ( ) y 64 x 4 =± x 9. ( ) ( ) x 6 + y+ 4 = 5. a. The radius of the circle is 5. b. The four points for the rough sketch of the graph are (1, 4), (11, 4), (6, 1), and (6, 9); the center of the circle is (6, 4). c. ( ) y 5 x 6 4 =± 10. ( ) ( ) x+ 1 + y+ = 81. a. The radius of the circle is 9. b. The four points for the rough sketch of the graph are ( 10, ), (8, ), ( 1, 7), and ( 1, 11); the center of the circle is ( 1, ). c. ( ) y 81 x 1 =± + y x

193 AIIF Page 188 SJ Page 93 For each of the Exercises 11 through 14, graph the given piece wise function on the provided grid. Also, graph the piece wise equation on your graphing calculator and write the format of the equation as it was entered into the graphing calculator for x < 0 y = 3x 1 for x 0 1. x+ 4 for x < 5 y = x/5 for x 5 Answers may vary. A sample response might Answers may vary. A sample response be: "I entered ( 1)(x<0) in Y1, (3x 1)(x 0) in Y." might be: "I entered (abs(x+4))(x< 5) in Y1, (x/5)(x 5) in Y." 13. ( x+ ) for x < y = x 4( ) for x 14. x > x y = ( x ) x 3 x < x Answers may vary. A sample response might Answers may vary. A sample response be: "I entered ( (x+)^)(x< ) in Y1, (4(^x))(x ) might be: "I entered ( x))(x < ) in Y1, in Y." ((x )^ ))( x and x ) in Y, (x^3)(x>) in Y3."

194 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 189 SJ Page A cell phone company offers broadband wireless internet access at a cost of $50 per month for the first 1 GB (giga byte) of usage. After the first 1 GB of usage, the company charges $0.50 per 1 MB (mega byte). Write a piece wise equation representing the total monthly cost for broadband wireless internet. Note: 1 GB = 1,000 MB. Label the independent variable and state what it represents. 50 for 0 x 1000 y = 0.05(x 1000) + 50 for x > 1000 The independent variable is x and it represents the amount of usage in megabytes ($0.05 for 1 MB). The dependent variable is y and it represents the total cost in dollars of broadband wireless internet usage x (MB)

195 AIIF Page 190 SJ Page 95 Outcome Sentences Graphing absolute value functions was similar to When graphing a circle equation on the graphing calculator When solving a circle equation for y Piece wise graphing was hard to understand because When graphing a piece wise equation I would like to know more about

196 Non Linear Functions Lesson 8: Miscellaneous Non-Linear Functions AIIF Page 191 Teacher Reference Lesson 8 Quiz Answers 1. Answers may vary. A sample table might be: x y A sample transformation response might be: "There is a horizontal translation of 1/ unit and vertical translation 3 units. The graph is contracted because of multiplication of x by.". a. The circle equation in general form is ( ) ( ) x y 4 = 49. b. The radius of the circle is 7 and the center is (-3, 4). c. The coordinates of the four points are ( 3, 11), ( 3, 3), (4, 4),and ( 10, 4). 3. Answers may vary. A sample response might be: "I entered ((x )^)(x ) in Y1 and ( x +6)(x > ) in Y."