Algebra II Foundations

Transcription

1 AIIF Algebra II Foundations Non Linear Functions Student Journal

2 Table of Contents Lesson Page Lesson 1: Introduction to Quadratic Functions...1 Lesson : The Quadratic Formula...8 Lesson 3: Graphing Quadratic Functions and Their Applications...15 Lesson 4: Power Functions...7 Lesson 5: Inverse Variation...39 Lesson 6: Eponential Functions...51 Lesson 7: Step Functions...67 Lesson 8: Miscellaneous Non Linear Functions...78 CREDITS Author: Contributors: Graphic Design: Dennis Goette and Dann Jones Robert Balfanz, Doroth Barr, Leonard Bequiraj, Stan Bogart, Robert Bosco, Carlos Burke, Lorenzo Haward, Vicki Hill, Winnie Horan, Donald Johnson, Ka Johnson, Karen Kelleher, Kwan Lange, Dennis Leah, Song-Yi Lee, Hsin-Jung Lin, Gu Lucas, Ira Lunsk, Sandra McLean, Hemant Mishra, Glenn Moore, Linda Muskauski, Trac Morrison, Jennifer Prescott, Gerald Porter, Steve Rigefsk, Ken Rucker, Stephanie Sawer, Dawne Spangler, Fred Vincent, Maria Waltemeer, Tedd Wieland Gregg M. Howell Copright 009, The Johns Hopkins Universit, on behalf of the Center for Social Organization of Schools. All Rights Reserved. CENTER FOR SOCIAL ORGANIZATION OF SCHOOLS Johns Hopkins Universit 3003 N. Charles Street Suite 00 Baltimore, MD fa All rights reserved. Student assessments, Cutout objects, and transparencies ma be duplicated for classroom use onl; the number is not to eceed the number of students in each class. No other part of this document ma be reproduced, in an form or b an 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 validit and content as it relates to the manual s corresponding topic. The Johns Hopkins Universit, and its licensors is not responsible for an changes in content, IP addresses changes, pop advertisements, or redirects. It is further recommended that teachers confirm the validit of the listed addresses if the intend to share an address with students.

3 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 1 Lesson 1: Introduction to Quadratic Functions Activit 1 For the following eercises, write a matching quadratic equation then solve the equation and answer each question. 1. The square of a number is sit-four. What are the numbers that make the equation true?. The square of the difference of a number and si is one hundred twent-one. What are the numbers? 3. 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? Crop circles are patterns created b the flattening of 4. The area of a crop circle is approimatel 157 square feet. crops such as wheat, barle, Approimatel what is the size of the diameter of the circle? Use rapeseed, re, corn, linseed, 3.14 forπ. and so into circles. The term was first used b researcher Colin Andrews to describe simple circles he was 5. A compan s cost can be determined b the sum of the variable costs and the fied costs. Namel, C() = variable costs + fied researching. costs. It has been determined that the compan 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 compan's fied costs are $1,000.00, write an equation, in function notation, that represents the total costs of producing units of the product. b. If total costs are $,751,000.00, how man units of the product are produced? 6. A compan makes a product. The compan has determined the approimate cost to produce a single unit of the product. The compan has fied costs of $500. The compan 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 = c. Write a cost function, C(), which represents the total cost of producing units of the product. Use the information given to determine the cost to produce one unit of the product.

4 AIIF Page Activit In this activit, ou will investigate the graphs of various quadratic equations. For the following eercises, find the coordinates of the minimum or maimum value and state the minimum or maimum value, all intercepts and intercepts, and make a sketch of the graph in the grid provided. The eercises are set up in most cases to draw two graphs per grid. You ma also want to displa the two graphs simultaneousl on our graphing calculator as well. NOTE: Displaing table values on our graphing calculator ma help ou to draw the graph =. = 4 3. = =

5 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 3 = ( ) = 6. ( ) = ( ) 8. ( ) = 3

6 AIIF Page 4 9. For the given graph, identif the coordinates of the minimum point, intercepts, intercept, and equation representing the graph. 10. From Eercises 1 through 9, what conclusions and characteristics can ou make about the graphs of quadratic equations?

7 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF Page 5 Practice Eercises Solve each of the following. 1. The square of a number is one hundred fort-four. Write a quadratic equation and then solve for the unknown number(s).. The square of the sum of a number and nine is one hundred sit-nine. Write a quadratic equation and then solve for the unknown number(s). 3. The Sparkling Diamonds jewelr 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 b the equation P= C =, where P is the profit and C is the cost of the item? Graph the quadratic equation = on the grid supplied below. Label all intercepts and determine the maimum or minimum point. 5. Graph the quadratic equation ( 4) determine the maimum or minimum point. = on the grid supplied below. Label all intercepts and 4. 5.

8 AIIF Page 6 6. Graph the quadratic equation ( ) minimum point. = Label all intercepts and determine the maimum or 7. For the given graph, identif the minimum point, intercepts, intercept, and equation representing the graph. 8. How does the process of squaring relate to quadratic functions?

9 Non Linear Functions Lesson 1: Introduction to Quadratic Functions AIIF 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 maimum when The minimum or maimum of a quadratic equation can be determined b I can use the graphing calculator to I would like to find out more about I now understand I still have a question about

10 AIIF Page 8 Lesson : The Quadratic Formula Activit 1 In this activit, ou will be solving quadratic equations using the quadratic formula to find the values of when =0. Make sure the equations are written in general form before determining the coefficients a, b, and c. Standard Form: General Form: = a + b + c a b c + + = 0 For the following eercises: a. Write the quadratic equation in general form. b. Identif the values of a, b, and c. c. State the nature of the roots b calculating the discriminant. d. Find all solutions, if an, for the quadratic equation. Quadratic Formula: ± = b b 4ac a 1. = = = = =

11 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 9 6. For the given quadratic formula, identif the values a, b, and c and write the matching quadratic equation in standard form. Note: use = a + b + c. a. 5± 5 4(1)(3) = (1) b. 7 ± ( 7) 4(3)( 4) = (3) c. 8 ± ( 8) 4( 9)(1) = ( 9)

12 AIIF Page 10 Activit In this activit, our teacher will guide ou through writing a program for the quadratic formula on the classroom graphing calculator. Test our program on the first two activit. Things ou will need to pa attention to in our program are: The discriminant Programming logic Data input Data output Calculations using the quadratic formula Use the supplemental eercises below to further test our program b solving for when =0. Round our answers to 3 decimal places. 1. = = = = = Can ou think of an improvements in the program ou wrote?.

13 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 11 Activit 3 In this activit, ou will continue to use the quadratic formula to solve quadratic equations for real-world applications. Use the same process from Activit 1 to find our solutions (write the equation in general form; identif the coefficients a, b, and c.) Make sure our answers make sense for the real-world application problem. Break Even Point The point where the revenue, R(), equals the cost, C(). Smbolicall, R() = C(). 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 b the functionc ( ) = and the owner knows he can sell the product for $35 each, giving him a total revenue of R ( ) = 35, where represents the number of items produced. The owner would like ou to find the break-even points so he can determine the number of product items he should produce each week.. 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 b 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? 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 b the equation st ( ) = 4.09t t. a. How far has the Corvette traveled after 4.3 seconds, to the nearest foot? b. How long does it take the Corvette to travel half a mile? Round our answer to the nearest tenth of a second. Note: A mile is 5,80 feet. 4. The Coast Guard is testing two rescue flares from two competing companies. The Coast Guard plans to sign a contract with the compan 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 b: Compan A: = Compan B: = where is the height and is the horizontal distance traveled. Determine which flare the Coast Guard should purchase b substituting = 0 into each equation and finding. What does the constant 15 represent in each equation?

14 AIIF Page 1 Practice Eercises For Eercises 1 through 3: a. Write the quadratic equation in general form. b. Identif the values of a, b, and c. c. State the nature of the roots b calculating the discriminant. d. Find all solutions, if an, for when =0 for the quadratic equation. Round all answers to the nearest tenth. 1. = = = Co s formula for measuring velocit of water draining from a reservoir through a horizontal pipe is 100HD = 4v + 5v, where v represents the velocit L 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 our answer to the nearest tenth of a foot per second. H L D

15 Non Linear Functions Lesson : The Quadratic Formula AIIF Page 13 SJ Page # 5. A ball is thrown upward with an initial velocit 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 velocit, and h 0 is the initial height. Round our answer to the nearest tenth of a second. 6. For the given quadratic formula, identif the values a, b, and c and write the quadratic equation from these values. a. 11 ± (11) 4(5)(6) = (5) b. 1 ± ( 1) 4( )( 19) = ( ) c. For part b. above, will the quadratic equation have an real solutions? Eplain. 7. Find the mistake below and correct it. 13= 7 ( 13) ± ( 13) 4(1)(7) = (1) 13 ± = 13 ± 141 = 13 ± = 1.45 and = 0.55

16 AIIF 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

17 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 15 Lesson 3: Graphing Quadratic Functions and Their Applications Activit 1 In this activit, ou will use our knowledge of the vertical line of smmetr to plot points, draw a graph, and find the equation for the vertical line of smmetr. 1. Using the given dashed vertical line of smmetr, plot and draw the missing half of the graph. Write the equation for the vertical line of smmetr. State whether the graphs have a minimum or a maimum value and eplain wh. a. b.. Complete the tables below using the values in the table along with the equation for the vertical line of smmetr. Plot the points in the table, draw the graph, and draw the vertical line of smmetr. State the equation of the vertical line of smmetr and whether the data tables have a minimum or a maimum value and eplain wh. a. Vertical line of smmetr: b. Vertical line of smmetr: 1/ 5 1 1/ 5/ /

18 AIIF Page Write the equation for the vertical line of smmetr for the given graphs. State whether the graphs have a minimum or a maimum value and eplain wh. a. b. 4. Write the equation for the vertical line of smmetr for the data tables below. State whether the data tables have a minimum or a maimum value and eplain wh. Also state the minimum or maimum value. a. b

19 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications Activit In this activit, ou 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 eercises ou will need to: a. Identif the values of a, b, and c b. Verte coordinates c. All intercepts d. Line of smmetr e. Several points on either side of the verte b Verte coordinate: = a b Verte coordinate: = f a b Line of Smmetr: = a Quadratic Formula: Discriminant: b 4ac ± = AIIF b b 4ac a Page = = = + 5

20 AIIF Page Photosnthesis is the process in which plants use the energ from the sun's ras to convert carbon dioide to ogen. The intensit of light is measured in lumens. Let R be the rate that a certain plant uses to convert the sun's light energ. Let be the intensit of the light. The plant converts the carbon dioide at a rate according to the equation R= Sketch the graph of this equation and determine the intensit that gives the maimum rate of photosnthesis. State the domain which makes sense for the application. 5. The cost function to make a certain product is C ( ) = The revenue function for the same product is given b R ( ) = a. Graph the cost and revenue cost functions on the same set of aes. b. What level of production will produce the maimum revenue? What is the maimum revenue? c. What level of production will produce the minimum cost? What is the minimum cost? d. Graph the profit function (profit = revenue minus cost) on a separate set of aes. e. What level of production will produce the maimum profit? What is the maimum profit?

21 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page The graph below represents the profit function for a compan that produces widgets. Find the equation of profit function P(). Note: You should be able to determine the value of c from the graph. Also, use the coordinates of the verte to find a and b. Use b/a for the coordinate and solve for b in terms of a and substitute this value into = a + b + c to find a and then b A town is having a parade and celebration for its high school marching band. The school s marching band recentl marched in Mac s Annual Thanksgiving Da 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?

22 AIIF Page 0 Activit 3 In this activit, ou will use our knowledge and understanding of quadratic functions to do quadratic regression on scatter plots and data sets. In the last activit, ou wrote quadratic functions from graphs. In the Linear Functions unit, ou drew the best fit line for a scatter plot and determined the equation for the line of best fit. In this activit, ou 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 ou drew. 1. Which scatter plots below seem to have a quadratic trend? A B C D. Draw a best fit parabola for the scatter plots ou determined had a quadratic trend in Eercise Determine the quadratic functions from the best fit parabolas ou drew in Eercise.

23 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 1 In the Linear Functions unit, ou 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 our 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 b age () and percentage (). Under 5 5 to to to and over % 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. b. Use our graphing calculator to create a scatter plot and graph of the data and sketch the scatter plot and graph on the set of aes. c. How well does the graph of the best fit parabola fit the data? 5. The students of Mr. G's class were told to record the number of hours spent studing for their mathematics test. For each student, Mr. G wrote an ordered pair (, ). The -value represented the number of hours the student spent studing and the -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 our graphing calculator to create a scatter plot. Does the data seem to model a quadratic equation? Eplain.

24 AIIF Page b. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three decimal places. c. Use our graphing calculator to create a scatter plot and graph of the data on the same set of aes. d. How well does the graph of the best fit parabola fit the data? 6. The table below is the U. S. Census (in millions of people) for the ears 1810 through 000. The -values represent the ear the Census was taken and the -values represent the population in millions of people. Note: = 0 for the ear 1810, = 10 for the ear 180, etc a. Use our graphing calculator to create a scatter plot. Does the data seem to model a quadratic equation? Eplain. b. Determine the equation for the parabola of best fit. Round the values of a, b, and c to three decimal places. c. Using our equation of best fit, predict the population for the Census in 010 and 00.

25 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 3 Practice Eercises Graph Eercises 1 and 3. Make sure to include the following: a. Identif the values of a, b, and c. b. Verte coordinates. c. All intercepts. d. Line of smmetr. e. Several points on either side of the verte. NOTE: Round answers to nearest tenth. 1. = =

26 AIIF Page 4 3. A ball is thrown directl upward from an initial height of 00 feet with an initial velocit of 96 feet per second. After 3 seconds it will reach a maimum height of 344 feet. The standard form of a quadratic equation for a projectile is given b st () = 16t + vt 0 + s 0, where s(t) is the projectiles height at time t, v 0 is the initial velocit, and s 0 is the initial height. What is the equation of the quadratic function for this problem? What does the intercept represent? Graph the quadratic function. Round answers to nearest tenth if necessar. 4. Suppose that in a monopol market (a market with a downward sloping curve) the total cost per week of producing a particular product is given b the cost function C ( ) = The weekl demand for the product is such that the revenue function is R ( ) = Graph both functions on the same set of aes and shade the region that represents the area in which the compan is making a profit. Find the points of intersection for the cost and revenue functions. What do the points of intersection represent?

27 Non Linear Functions Lesson 3: Graphing Quadratic Functions and Their Applications AIIF Page 5 5. Determine the quadratic function from the graph at the right. 6. A ball was dropped from a height of approimatel 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. b. How long does it take for the ball to hit the ground? Round our answer to the nearest hundredth of a second. HINT: Use the quadratic formula. 7. You run a biccle rental business for tourists during the summer in our town. You charge $10 per bike and average 0 rentals a da. An industr 300 journal sas that, for ever 50 cent increase in rental price, the average 00 business can epect to lose two rentals a da. The graph to the right represents the quadratic equation 100 used to determine how man, if an, 50 cent increases are needed to maimize revenue. Let represent the number of increases to the current charge rate. Negative values for represent 50 cent decreases. Use this information and the graph to find the quadratic equation to maimize revenue. What should ou charge per bike rental? What is our maimum profit?

28 AIIF Page 6 Outcome Sentences The verte is determined b The line of smmetr is used for Applications of quadratic equations reall help me to understand Quadratic functions and applications of quadratic functions are graphed b The easiest wa for me to write a quadratic equation from a graph is b The most difficult part of graphing is because Quadratic modeling with the graphing calculator

29 Non Linear Functions Lesson 4: Power Functions AIIF Page 7 Lesson 4: Power Functions Activit 1 In this activit, ou will write and solve power tpe equations and their applications. Let's look at the following application problem: The area of a cube shaped bo is 64 cubic inches. What are the dimensions of the bo? What are the steps necessar to setup and solve problems of this tpe? We need to start b labeling the known and unknown (variable) information. Net, 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 Eercises 1 through 3, write and solve a power tpe equation. 1. The cube of a number is 15. What is the number?. Si is added to a number that was raised to the sith power. If the sum is 735, what was the number that was raised to the sith power? 3. The difference of a number and si, raised to the fourth power, is 56. What are the numbers? 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 is V = π r where r is the radius. Use 3.14 for 3 the value of π. 5. A couple plans to invest $5,000 into an account that is compounded annuall for 5 ears. The hope to have $75, after the 5 ears. What interest rate will guarantee that their investment of $5,000 will grow to $75, after the 5 ears? 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 ears invested.

30 AIIF Page 8 Activit In this activit, ou will investigate the graphs of power and power like functions. In our description include whether the graph is an even or odd function. Part A: How do different powers affect the graph of =? n Function n = Describe or Draw General Shape Describe location of maimum or minimum Describe similarit or difference to the graph of = = 3 = 4 = 5 = 6 = 7 Write our overall conclusion as to how different powers affect the graph of =. n

31 Non Linear Functions Lesson 4: Power Functions AIIF Page 9 Part B: How do different coefficients affect the graph of = a? n Function n = a Describe or Draw General Shape Describe location of maimum or minimum Describe similarit or difference to the graphs 3 of = or = = = 4 = 9 = 1 = 3 = 5 3 = 5 3 = 1 3 Write our overall conclusion as to how different coefficients affect the graph of = a. n

32 AIIF Page 30 Part C: How does adding or subtracting a constant, k, to = affect the graph of the equation? n Function n = ± k Describe or Draw General Shape Describe location of maimum or minimum Describe similarit or difference to the graphs 3 of = or = = + 1 = + 3 = = 4 3 = = = 6 Write our overall conclusion as to how adding or subtracting different constants affect the graph of =. n

33 Non Linear Functions Lesson 4: Power Functions AIIF Page 31 Part D: How does adding or subtracting a constant, h, to the value before completing the power, in the n equation =, affect the graph? Function = ( ± h) n Describe or Draw General Shape Describe location of maimum or minimum Describe similarit or difference to the graphs 3 of = or = = ( + 1) ( 3 ) = + ( ) = ( 4 ) 3 = + ( 6 ) 3 = + ( 3 ) 3 = Write our overall conclusion to how adding or subtracting a constant to the value in the equation affected the graph. = n Write our overall conclusion as to what affect the values of a, h, k, and n have on the graph of = a ( ± h) n ± k.

34 AIIF Page 3 For Eercises 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 maimum or minimum value. If it does, state the value of the maimum or minimum. c. If the function, represented b the graph, is a power function then determine if it is even, odd, or neither

35 Non Linear Functions Lesson 4: Power Functions AIIF Page Match the equation with its graph. a. = 4 b. = 3 c. = 3 A. B. C.

36 AIIF Page 34 For Eercises 6 through 10, state an vertical or horizontal translation from the first equation to the second. Sketch a rough graph of the equations showing translation (do not worr about scale). 6. = 3 and = = 4 and = = 5 and = ( 6) = 5 and = ( 6) = 6 and = ( + 1)

37 Non Linear Functions Lesson 4: Power Functions AIIF Page 35 Practice Eercises For the Eercises 1 through 3, write and solve a power like equation. 1. The fifth power of a number is 43. What is the number?. 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? 3. The sum of a number and three, raised to the third power, is What is the number? 4. The volume of a cubic bo is approimatel 151 cubic inches. What are the lengths of the sides of the cubic bo? Round our answer to the nearest tenth of an inch. 5. Darnell and Shanice plan to invest $50, into an account that is compounded annuall at a rate of 3.5%. Create a table of values that represents what their investment is worth after 4, 8, 1, and 16 ears. 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 ears invested. Round the value of the investment to the nearest cent. Years Invested (t) Value of Investment in dollars (S) 6. Darnell and Shanice plan to use the total value of the investment in 16 ears for a college education for their onl child. Approimatel how much will the have available each ear, for four ears, for their child's education? Round our answer to the nearest thousand dollars.

38 AIIF Page 36 For Eercises 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 maimum or minimum value and state the value of the maimum or minimum. c. State if the function is even, odd, or neither. d. State an vertical or horizontal translation from the origin. e. Sketch a rough graph of the power or power like function. 7. = 4 8. = ( + 3) 3 9. = ( ) 5 4

39 Non Linear Functions Lesson 4: Power Functions AIIF Page Determine the power like function from the given graph. 5 (5, 7) (, 0) ( 1, 7) 5

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

41 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 39 Lesson 5: Inverse Variation Activit 1 In this activit, ou will create a bar graph that represents the equation = 1/, for values 1 through Cut out the grid template. Obtain a piece of construction paper from our teacher and cut a strip that is 1 centimeter wide b 180 millimeters long. Notice the 1 unit location on the vertical ais 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 ais. 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 Eercise 1. Place the cut strips along the horizontal ais 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 -values from the table and then draw a smooth curve connecting the points on our scatter plot. = 1/ Decimal Value

42 AIIF Page What do ou notice about the values of, which represent the lengths of the strips, as the values of increase? 6. As the values of get larger and larger, what value does seem to approach? 7. Using our graphing calculator, determine the values of - for each value in the table below. Write the values in the right column of the table. = 1/ 1/ 1/5 1/10 1/50 1/100 1/500 1/1000 1/ / / From our results from Eercise 7, as gets closer to 0 what value does get closer to? 9. Can we find the value of for = 0? Eplain. 10. Investigate variations of the inverse function b using our graphing calculator. a. Graph = /. Describe the differences between this graph and the graph = 1/. b. Graph = 3/. Describe the differences between this graph and the graph = 1/. c. Graph = 1/. Describe the differences between this graph and the graph = 1/. d. Graph = /. Describe the differences between this graph and the graph = 1/.

43 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 41 Activit 1 Grid Template 1 unit Cut Here

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45 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 43 Activit 1 Strip Cutouts Cut Here

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47 Non Linear Functions Lesson 5: Inverse Variation AIIF Page 45 Activit In this activit, ou will solve inverse variation problems and real world inverse variation problems. The general form for an inverse variation equation is = k/, where k is called the constant of proportionalit. Another wa of writing this equation is = k. You will investigate inverse variation and k b using the = k equation. 1. Find five different sets of values (ordered pairs) that make = 4 true. a. As the values of increase what do ou notice about the values of? b. Wh would the values of have to decrease as increases to keep the equation true? c. If the value doubles what happens to the value? d. If the value triples what happens to the value? e. What happens to the relationship between and if we change the constant to a different number such as 36? f. Wh do ou think equations in the form of =k, where k is constant, are called inverse variation equations? For Eercises and 3, use the given information to solve for the constant of proportionalit k and then for the unknown value of.. If varies inversel with and = 34 when = 1/68, what is the value of when =? 3. If varies inversel with the cube of and = 10 when = 4, what is the value of when =?

48 AIIF Page 46 For eercises 4 through 7, use the information given in the problem to find the constant of proportionalit k and answer the question. 4. The number of hours, h, it takes to mow a lawn varies inversel 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? b. Write an inverse variation equation for the problem. 5. Bole's law states that in a perfect gas where mass and temperature are kept constant, the volume, V, of the gas will var inversel 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? b. Write an inverse variation equation for Bole's law. 6. In hdraulics, the fluid pressure, P in pounds per square inch, is related directl with the force, f in pounds, and inversel 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? b. Write an inverse variation equation for the fluid pressure.

49 Non Linear Functions Lesson 5: Inverse Variation AIIF Page The weight of a bod varies inversel as the square of its distance from the center of the Earth. 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? b. Write an inverse variation equation for the weight of a bod. 8. Use the graph to the right, to write an inverse variation equation

50 AIIF Page 48 Practice Eercises For Eercises 1 through 3, use the given information to solve for the constant of proportionalit k and then for the unknown value. Write an inverse variation equation for each eercise. 1. If s varies inversel with t and s = 30 when t = 30, what is the value of s when t = 10?. If varies inversel with the fourth power of and = when = 3, what is the value of when = 0.1? 3. If j varies inversel with the square of l and j = 16 when l = 4, what is the value of j when l = 8? 4. The current, I in amps, produced b a batter varies inversel to the resistance, R in ohms, of the circuit to which the batter 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? b. Write an inverse variation equation for the current of the batter. 5. The intensit, I, of light observed from a source of constant luminosit varies inversel as the square of the distance, d, from the object. a. If the intensit of a light is lumens when the light source is 1.1 meters awa, what is the intensit of the light if the source is 3 meters awa? Round all answers to four decimal places. b. Write an inverse variation equation for the intensit of light, I, a distance d from the source.

51 Non Linear Functions Lesson 5: Inverse Variation AIIF Page Lengths of radio waves var inversel with radio wave's frequenc. a. Radio station WJHU broadcasts their FM signal with a frequenc of 88.1 MHz and has a wavelength of approimatel 3.4 meters. Boston's famous WRKO AM radio station broadcasts their signals with a frequenc of MHz. What is the wavelength of WRKO's broadcasts? NOTE: Round our k value to the nearest whole number and the wavelength to the nearest tenth of a meter. b. Write an inverse variation equation for the wavelength of radio waves.

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

53 Non-Linear Functions Lesson 6: Eponential Functions AIIF Page 51 Lesson 6: Eponential Functions Activit 1 Bacterial Growth Respirator Sstem Model Respirator sicknesses (infections), such as bronchitis and pneumonia, are caused b bacteria. Once bacteria gets in our lungs, the can duplicate (reproduce) at a certain rate. The following eperiment will model the amount of bacteria present over time. In this eperimental model, we will use small construction paper squares of one color to represent the bacteria. Eperiment Step 1: Cut out 64 red construction paper squares. Each square should be the same size and shape. The best size is 1 inch b 1 inch or 1 centimeter b 1 centimeter. Use a ruler to draw the squares before cutting. Eperiment Step : Cut out the lungs template at the end of the activit. Eperiment Step 3: Place one red square on the lung template (an where inside the lung area.) This represents the initial amount of bacteria, a single cell. Note: Bacteria are actuall ver small in size. A single cell of bacteria is about 1/10,000 th of a centimeter. Eperiment Step 4: Ever minute, add enough red squares to double the amount ou had previousl. This represents the bacteria duplicating (reproducing itself) ever hour. While ou are waiting for each minute to end, count out the necessar squares that ou will be adding for the net minute. Also, record the time and amount of bacteria present in the lungs in the table provided below. Eperiment Step 5: Repeat Step 4 until all 64 squares have been placed "in" our lungs. Eperiment Step 6: You should realize that our table matches the table from the Setting the Stage transparenc. Table 1: Bacterial Growth Eperiment Hour Bacteria Count

54 AIIF Page 5 1. Create a scatter plot of the hours compared to the number of bacteria in the lungs. What tpe of pattern occurred in the scatter plot graph?. What is the rate that the bacteria are growing? 3. Graph a scatter plot of our data on a graphing calculator. Set the window range to an minimum of, maimum of 7, scale of 1, minimum of, maimum of 100, and scale of 10. Is the scatter plot linear? If not describe the shape of the graph. Bacteria Count Bacteria Growth Eperiment Hour 4. How man bacteria do ou epect to be in the lungs after a 4 hour period? How might ou calculate this value? 5. Approimatel how man hours will it take until there are 1 trillion (1,000,000,000,000 or ) bacteria in the lungs? NOTE: The graphing calculator ma displa 1 trillion as 1.0 E1.

55 Non-Linear Functions Lesson 6: Eponential Functions AIIF Page 53 Antibiotic Deca in the Blood Stream Eperimental Model To help cure illnesses antibiotics and/or medicines taken into the bod are circulated throughout the bod b the bloodstream. The kidnes take the drugs out of the blood. We saw, from the first part of the activit, 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 onl wa to become health again is b the use of antibiotics. The following eperiment will model the amount of antibiotics left in the bloodstream over time. In this eperimental 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. Eperiment Step 1: Eperiment Step : Eperiment 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 b 1 inch. Use a ruler to draw the squares before cutting. Place 0 red squares and 0 blue squares in a container (bag or bo). 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 quickl that represents the wa drugs leave the bloodstream. Shake the container and randoml remove 10 squares. Replace them with 10 red squares. Determine how man blood squares and antibiotic squares are now in the container. Place this information in Table 1 below. This step models the kidnes randoml cleaning one quarter of the blood each hour. Eperiment Step 4: Repeat Step 3 ten times. Place the information for each cleaning ccle in Table, Antibiotics Deca Eperiment, below. Table : Antibiotics Deca Eperiment Hour Blood Count Antibiotic Count

56 AIIF Page Create a scatter plot of the hours compared to the number of antibiotics left in the bloodstream. What tpe of pattern occurred in the scatter plot graph? 7. Create a transparenc cop of our graph. Place all the transparencies from each group on the overhead at one time and line up the aes. What do ou notice about the graph? Antibiotic Count Antibiotic Deca Eperiment Hour 8. If no new antibiotics are added, what would the graph do if we continued with the eperiment? 9. Graph a scatter plot of our data on a graphing calculator. Set the window range to an minimum of, a maimum of 1, a minimum of, and a maimum of 4. Is the scatter plot linear? If not describe the shape of the graph. 10. Graph = 0(0.75) on the same graph as the scatter plot. Describe how the graph of = 0(0.75) fits the data from the scatter plot.

57 Non-Linear Functions Lesson 6: Eponential Functions AIIF Page 55 Lungs Template Cut Here

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59 Non-Linear Functions Lesson 6: Eponential Functions AIIF Page 57 Activit In this activit, ou will determine the intercept, determine the tpe of graph, and draw a rough sketch of eponential functions. For Eercises 1 through 4: a. Determine the coordinates of the intercept b. Tpe of graph: growth or deca c. Draw a rough sketch of the eponential function on the grid provided. Note: set grid scale appropriatel = = = = 7(4 )

60 AIIF Page 58 For Eercises 5 through 7, state the intercept and the tpe of graph

61 Non-Linear Functions Lesson 6: Eponential Functions AIIF Page 59 Activit 3 In this activit, ou will solve real world eponential problems. 1. Your grandparents put $10,000 in an investment account, which collects interest four times a ear, when ou were born for our college education. The future value of our college education fund can be determined b the 4 function S = 10000(1.0375) t, where t represents the number of ears for the investment. How much mone will ou have available when ou start college? Assume ou will be 18 ears old when ou start college. Draw a rough sketch of the investment; set ais scales accordingl Viruses can produce man more offspring than bacteria per infection. Some viruses produce at an t eponential 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 man viruses will be present after 4 hours if there initiall were 5 viruses and the viruses produce a new generation ever 4 hours? 3. It has been determined that a certain cit has been growing eponentiall over the last 0 ears 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 ears that the town has been increasing. If the town initiall had 450 people 0 ears ago and the now have 1,443 people, what was the rate of increase in population over the last 0 ears? Round our answer to the nearest whole percent. 4. A local retail store has determined that its sales could grow eponentiall based on the amount the spend on advertising each week b 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 the advertised. If the store averaged 15 sales per week before advertising began, how man sales can the epect to have, each week, after advertising for 4 consecutive weeks? Round our answer down to the nearest whole sale.

62 AIIF Page The radio active deca of a material is given b the function ( t/t A = A ) 0e, where A 0 is the initial amount of the material, t is the amount of time in ears, and T is the half life of the radio active material. Plutonium 40 has a half life of 6540 ears. If a nuclear power plant started with 100 pounds of Plutonium 40, how much would be left after 0 ears? How man ounces of plutonium decaed during the 0 ears? Round our answers to the nearest hundredth pound and ounce.

63 Non-Linear Functions Lesson 6: Eponential Functions AIIF Page 61 Activit 4 In this activit, ou will use eponential regression to obtain an eponential function from real world data. 1. The following data table represents the dail 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 tpe of graph does the data model? b. What is the eponential regression function? Round values to three decimal places. c. How man commuters would ou epect if the had to pa $75 each da in commuting epenses? Round our answer to the nearest commuter.. The following data table represents the population of the United States from the ears 1790 through 000, where ear 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 tpe of graph does the data model? b. What is the eponential regression function? Round values to four decimal places. c. Using this eponential equation, what might ou predict will be the size of the U. S. population in the ear 060? Round our answer to the nearest ten thousandths. Note: Remember our current units for population is in millions.

64 AIIF Page 6 3. The following table represents the earl 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 tpe of graph does the data model? b. What is the eponential regression function? Round values to three decimal places. c. The U. S. oil production peaked in What could ou predict was our countr's peak output of oil in 1970? Round our answer to the nearest whole barrel. d. The actual U. S. oil production in 1970 was approimatel 3,500,000,000 barrels. What can ou sa about our predicted value of production compared to the actual value of production? e. What suggestion would ou make on limiting the use of our eponential regression function?