ALGEBRA 2 NY STATE COMMON CORE

Transcription

1 ALGEBRA NY STATE COMMON CORE Kingston High School emathinstruction, RED HOOK, NY 1571, 015

2 Table of Contents U n i t 1 - Foundations of Algebra... 1 U n i t - Linear Functions, Equations, and their Algebra U n i t - Eponential Functions U n i t 4 - Logarithmic Functions U n i t 5 - Sequences and Series U n i t 6 - Quadratic Functions and Their Algebra U n i t 7 - Graphic Characteristics of Functions U n i t 8 - Etension Lessons for Honors course... 6 U n i t 9 - Regression U n i t Polnomial and Rational Functions U n i t The Circular Functions U n i t 1 - Probabilit... 9 U n i t 1 - Statistics... 44

3 A l g e b r a U n i t 1 - Algebraic Essentials Review U n i t 1 - Foundations of Algebra LESSON 1: VARIABLES, TERMS, AND EXPRESSIONS Math has a unique language to clarif concepts and remove ambiguit from the analsis of problems. Here are some basic definitions so that we can all speak this language. Algebra II starts with some review of Algebra I. SOME BASIC DEFINITIONS Variable: A quantit that is unknown, or can change within the contet of a problem, represented b a letter or smbol. Terms: A single number or combination of numbers and variables using onl multiplication or division. Epression: A combination of terms using addition and subtraction. Eercise #1: Consider the epression 7. (a) How man terms does this epression contain? (b) Evaluate this epression, without our calculator, when. Show our calculations. (c) What is the sum of this epression with the epression 5 1? LIKE TERMS Like Terms: Two or more terms that have the same variables raised to the same powers. Onl the coefficients (the multipling numbers) can differ. To add like terms, simpl add the coefficients and leave the variables and powers unchanged. But, wh does this work? Below is an eample of the technical steps to combine two like terms Eercise #: Because the epression can be rewritten into a simpler form 1 9, these two epressions are equivalent. How can ou test this equivalenc? Show work for our test. Unit 1 Lesson 1 Variables, terms, and Epressions Page 1

4 A l g e b r a U n i t 1 - Algebraic Essentials Review LESSON 1: VARIABLES, TERMS, AND EXPRESSIONS HOMEWORK 1. For each of the following epressions, state the number of terms. (a) 1 (b) 8 7 (c) Simplif each of the following epressions b combining like terms. Be careful to onl combine terms that have the same variables and powers. (a) (b) (c) 4 9 (d) Given the algebraic epression do the following: 1 (a) Evaluate the epression for when 7. (b) Evaluate the epression for when 4. (c) Nina believes that this epression is equivalent to dividing 1 b one less than. Do our results from (a) and (b) support this assertion? Eplain. Unit 1 Lesson 1 Variables, terms, and Epressions Page

5 A l g e b r a U n i t 1 - Algebraic Essentials Review HOMEWORK (cont.) 4. Classif each of the following as either a monomial (single term), a binomial (two terms) or a trinomial (three terms). (a) 4 (b) 1 (c) 16 (d) 5 (e) 5 5 (f) 16 10t 4t 5. Use the distributive propert first and then combine each of the following linear epressions into a single, equivalent binomial epression. (a) (b) Which of the following is equivalent to the epression (1) 8 () 4 () 5 1 (4) ? 7. Simplif the epression Unit 1 Lesson 1 Variables, terms, and Epressions Page

6 A l g e b r a U n i t 1 - Foundations of Algebra LESSON : SOLVING LINEAR EQUATIONS You will learn man new equation solving techniques in Algebra, but the most basic of all equations are those where the variable, sa, is onl raised to the first power. These are called linear equations. You need to easil solve linear equations in order to be successful in the beginning of Algebra. Let's practice. Eercise #1: Solve each of the following linear equations for the value of. (a) 5 6 (b) (c) 8 6 (d) Strange things can sometimes happen when solving linear (and other) equations. Sometimes we get no solutions at all, in which case the equation is known as inconsistent. Other times, an value of will solve the equation, in which case it is known as an identit. Eercise #: Tr to solve the following equation. State whether the equation is an identit or inconsistent. Eplain Unit 1 Lesson Solving Linear Equations Page 4

7 A l g e b r a U n i t 1 - Foundations of Algebra Eercise #: An identit is an equation that is true for all values of the substitution variable. Tring to solve them can lead to confusing situations. Consider the equation: 6 1 (a) Test the values of 5 and in this equation. Show that the are both solutions. (b) Attempt to solve the equation until ou are sure this is an identit. Eercise #4: Which of the following equations are identities, which are inconsistent, and which are neither? (b) (a) (c) 8 7 (d) 1 1 Unit 1 Lesson Solving Linear Equations Page 5

8 A l g e b r a U n i t 1 - Foundations of Algebra LESSON : SOLVING LINEAR EQUATIONS HOMEWORK 1. Solve each of the following linear equations. If the equation is inconsistent, state so. If the equation is an identit, also state so. Reduce an non-integer answers to reduced fractions. (a) 75 5 (b) 7 5 (c) (d) (e) 1 9 (f) (h) 5 (Cross multipl to begin) 6 18 (g) (i) (j) 18 7 Unit 1 Lesson Solving Linear Equations Page 6

9 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.). Laura is thinking of a number such that the sum of the number and five times two more than the number is 6 more than four times the number. Determine the number Laura is thinking of.. As if # wasn't confusing enough, Laura is now tring to come up with a number where three less than 8 times the number is equal to half of 16 times the number after it was increased b 1. No number seems to work. Eplain wh. 4. When finding the intersection of two lines, ou first "set the linear equations equal" to each other. Find the intersection point of the two lines whose equations are shown below. Be sure to find both the and coordinates. 5 1 and Eplain wh ou cannot find the intersection points of the two lines shown below. Give both an algebraic reason and a graphical reason. 4 1 and 4 10 Unit 1 Lesson Solving Linear Equations Page 7

10 A l g e b r a U n i t 1 - Foundations of Algebra LESSON : COMMON ALGEBRAIC EXPRESSIONS In Algebra ou will spend a lot of time evaluating and simplifing algebraic epressions. ALGEBRAIC EXPRESSION Algebraic epressions are just combinations of constants and variables using addition, subtraction, multiplication, and division along with eponents and roots (square roots, cube roots, etc.). You must be able to evaluate algebraic epressions for values of the variables in them. Eercise #1: Consider the algebraic epression (a) Describe the operations occurring within this epression and the order in which the occur (b) Evaluate this epression for the replacement value. Show each step in our calculation. Do not use a calculator. 4 Eercise #: Consider the more comple algebraic epression (known as a rational epression) 7. (a) Without using our calculator, find the value of this epression when. Reduce our answer to simplest terms. Show our steps. (b) If a student entered the following into their calculator, it would give them the incorrect answer. Wh? 4 / ^ 7 Epressions can contain operators such as square and cube roots and absolute value. Practice these below: Eercise #: Is the absolute value epression 8 equivalent to 10? How can ou check this? Unit 1 Lesson Common Algebraic Epressions Page 8

11 A l g e b r a U n i t 1 - Foundations of Algebra Eercise #4: Consider the algebraic epression 5, which contains a square root. (a) Evaluate this epression for. (b) Wh can ou not evaluate the epression for 1? (c) Ma thinks that the square root operation distributes over the subtraction. In other words, he thinks he can take the square root of each TERM, and believes the following equation is an identit: Show that this is not an identit. 5 5 Algebraic epressions can be complicated, but if ou consider order of operations (PEMDAS) and work generall from inside to outside then ou can evaluate an epression for replacement values. 8 Eercise #5: Consider the rather complicated epression 5 4. (a) What operation comes last in this epression? (b) Evaluate the epression for. Simplif it completel. Eercise #6: Which of the following is the value of (1) 1 () 18 () 4 (4) when 10? Unit 1 Lesson Common Algebraic Epressions Page 9

12 A l g e b r a U n i t 1 - Foundations of Algebra LESSON : COMMON ALGEBRAIC EXPRESSIONS HOMEWORK 1. Which of the following epressions has the greatest value when 5? Show how ou arrived at our choice A zero of an epression is a value of the input variable that results in the epression having a value of zero (catch and appropriate name). Is a zero of the quadratic epression shown below? Justif our es/no answer Which of the following is the value of the rational epression 1 1 (1) () 1 4 () 5 (4) when? 4. If 5 and then is (1) 1 7 () 1 () 9 (4) 7 19 Unit 1 Lesson Common Algebraic Epressions Page 10

13 A l g e b r a U n i t 1 - Foundations of Algebra 5. What is the value of 10 if? (1) 7 () () 5 (4) 17 HOMEWORK (cont.) 6. If then has a value of (1) 5 () 5 () 7 5 (4) 1 7. The revenue, in dollars, that emathinstruction makes off its videos depends on how man views the receive. If represents the number of views, in hundreds, then the profit can be found with the epression: How much revenue would the make if their videos were viewed 600 times? 8. Sam believes that the two epressions below are equivalent. Test values and see if ou can build evidence for or against this belief Unit 1 Lesson Common Algebraic Epressions Page 11

14 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 4: BASIC EXPONENT PROPERTIES Eponents represent repeated multiplication. The wa the combine, or don't combine, is dictated b this simple premise. The process is to multipl the numerical coefficients and add the powers Find the product of the monomials 5 and 4. Eplain wh the final eponent on the variable is 7. Eercise #1: Multipl the following monomials. 6 4 (a) 5 (b) 6 (c) (d) 4 Remember, monomials (or terms) can have more than one variable, just as long as the are all combined using multiplication and division onl. Multipling monomials that have more than one variable still just involves application of eponent laws. Eercise #: Find each of the following products, which involve monomials of multiple variables. 5 (a) 4 5 (b) (c) Unit 1 Lesson 4 Basic Eponent Properties Page 1

15 A l g e b r a U n i t 1 - Foundations of Algebra One of the ke skills needed this ear is factoring epressions, especiall factoring out a common factor. To build some skills, consider the following problem. Eercise #: Fill in the missing blank in each of the following equations involving a product such that the equation is then an identit. 5 8 (a) 6 (b) 1 4 (c) 0 4 The final skill we will review in this lesson is using the distributive propert of multiplication (and division) over addition (and subtraction). Eercise #4: Use the distributive propert to multipl the following monomials and polnomials. (a) 5 (b) 5 6 (c) 7 (d) (e) 4 4 Eercise #5: Similar to Eercise #, fill in the missing portion of each product so that the equation is an identit. (a) (b) (c) (d) 4 9 Unit 1 Lesson 4 Basic Eponent Properties Page 1

16 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 4: BASIC EXPONENT PROPERTIES HOMEWORK Multipl and 7. Find each of the following products of monomials. 4 (a) 10 5 (b) (c) 4 8 (d) (e) 4t 15t (f) (g) 1 (h) Fill in the missing portion of each product to make the equation an identit. 6 (a) 18 (b) (c) (d) 4 (e) (f) Use the distributive propert to write each of the following products as polnomials. (a) 45 (b) 510 (c) (d) 10 8 (e) (f) 8 5 Unit 1 Lesson 4 Basic Eponent Properties Page 14

17 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.) (#4 cont.: Use the distributive propert to write each of the following products as polnomials) (g) (h) 16t t t (i) 1 5. Fill in the missing part of each product in order to make the equation into an identit. (a) (b) 8 10 (c) 18t 45t 5 9 t (d) (e) (f) Another important eponent propert occurs when a monomial with an eponent is raised to another power. See if ou can come up with a general pattern. 6. Write each of the following out as etended products and then simplif. The first is done as an eample. (a) 6 (b) 5 (c) 4 4 (d) a b 7. So, what is the pattern? For positive integers a and b:? Unit 1 Lesson 4 Basic Eponent Properties Page 15

18 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 5: MULTIPLYING POLYNOMIALS Polnomials are epressions that are combinations of terms using addition and subtraction that can have onl constants and positive integer powers. For eample is a polnomial. This lesson covers multipling polnomials. The distributive propert will be used, also an area model will be shown. Let's start b multipling binomials. How man terms does a binomial have? Eercise #1: Find the product of ( - 4) and ( +1) using the distributive propert. Eercise #: Consider the product of with 5 (a) Find this product using the distributive propert twice. (b) Represent this product on the area model shown below. 5 Eercise #: Find the product of the binomial 4 propert or using the area model. with the trinomial 5 using the distributive Unit 1 Lesson 5 Multipling Polnomials Page 16

19 A l g e b r a U n i t 1 - Foundations of Algebra It is critical to understand that when two polnomials are multiplied then the result is equivalent to this product and this equivalence can be tested. Eercise #4: Consider the product of and 5. (a) Evaluate this product for 4. Show the work that leads to our result. (b) Find a trinomial that represents the product of these two binomials. (c) Evaluate the trinomial for 4. Is it equivalent to the answer ou found in (a)? (d) What is the value of the trinomial when? Can ou eplain wh this makes sense based on the two binomials? Eercise #5: The product of three binomials, just like the product of two, can be found with repeated applications of the distributive propert. (a) Find the product: 4 7. (b) For what three values of will the cubic polnomial that ou found in part (a) have a value of zero? What famous law is this known as? (c) Test one of the three values ou found in (b) to verif that it is a zero of the cubic polnomial. Unit 1 Lesson 5 Multipling Polnomials Page 17

20 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 5: MULTIPLYING POLYNOMIALS HOMEWORK 1. Multipl the following binomials and epress each product as an equivalent trinomial. Use an area model to help find our product, if necessar. (a) 5 8 (b) 7 (c) 5 (d) 4 10 (e) 15 4 (f) 1 9. Find each of the following products in equivalent form. (a) 5 (b) (c) 5 Unit 1 Lesson 5 Multipling Polnomials Page 18

21 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.). A square of an unknown side length inches has one side length increased b 4 inches and the other increased b 7 inches. (a) If the original square is shown below with side lengths marked as, label the second diagram to represent the new rectangle constructed b increasing the sides as described above. (b) Label each portion of the second diagram with their areas in terms of (when applicable). State the product of 4 and 7 as a trinomial below. (c) If the original square had a side length of inches, then what is the area of the second rectangle? Show how ou arrived at our answer. (d) Verif that the trinomial ou found in part (b) has the same value as (c) for. 4. Epression 8 4. (a) For what values of will this epression be equal to zero? Show how ou arrived at our answer. (b) Write this product as an equivalent trinomial. (c) Show that this trinomial is also equal to zero at the larger value of from part (a). Unit 1 Lesson 5 Multipling Polnomials Page 19

22 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 6: USING TABLES ON YOUR CALCULATOR The graphing calculator is an amazing device that can do man things. One helpful function is evaluating epressions for different input values. We will be using two tools on the calculator toda, STORE and TABLE. First let's use STORE. The STORE button on our calculator is located above the ON button (lower left). Eercise #1: Find the value of each of the following epressions b using the STORE feature on our calculator. (a) 7 for 5 (b) 6 5 for 10 (c) 7 0 for Sometimes the calculator gives useful information even when it cannot evaluate an epression. Eercise #: Consider the epression 6. What happens when ou use STORE to evaluate this epression for 5? Evaluate the epression b hand to eplain what the calculator is telling us. Eercise #: Multipl the binomials (a) Find their product in trinomial form. and 5. (b) Evaluate both the trinomial and the original product for. What do ou notice? (c) Use STORE to evaluate the trinomial from (a) for 5. Eplain wh the trinomial turns out to be this specific value at 5 (look at the original binomials). Unit 1 Lesson 6 Using Tables on our Calculator Page 0

23 A l g e b r a U n i t 1 - Foundations of Algebra STORE is helpful when ou are determining the value of an epression at one or two input values of. But, if ou want to know an epression's value for multiple inputs, then TABLES are a much better tool. On the graphing calculator, the TABLE button is on the upper right, ( nd > GRAPH.) Eercise #4: The epression 16 has an integer zero somewhere on the interval Use a TABLE to find the zero on this interval. Show the table. Table commands are good at proving two epressions are equivalent. This is helpful when ou've done several manipulations and ou want to be sure ou've produced an algebraicall equivalent epression. Eercise #5: Consider the more comple algebraic epression shown below: 5 8 (a) This relativel comple epression simplifies into a linear binomial epression. Determine this epression carefull. Show our work below. (b) Set up a table using the original epression and the one ou found in (a) over the interval 0 5. Compare values to determine if ou correctl simplified the original epression Unit 1 Lesson 6 Using Tables on our Calculator Page 1

24 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 6: USING TABLES ON YOUR CALCULATOR HOMEWORK 1. Use the STORE feature on our calculator to evaluate each of the following. No work needs to be shown. (a) 7 18 for 8 (b) 5 for (c) for 5 (d) 5 8 for 1 (e) 4 5 for (f) 4 9 for 5. The STORE feature helps when checking to see if a value is a solution to an equation. Let's see how this works in this problem. Consider the linear equation: (a) Solve this equation for (b) Using STORE, determine the value of both the left hand epression, 6, and the right hand epression, 4 9, at the value of ou found in (a). (c) Wh does what ou found in part (b) verif that our solution is correct (or possibl incorrect if ou made a mistake in (a))?. Two of the following values of are solutions to the equation: are and provide a justification for our answer Determine which the Unit 1 Lesson 6 Using Tables on our Calculator Page

25 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.) 4. The quadratic epression 8 10 has its smallest value for some integer value of on the interval Set up a TABLE to find the smallest value of the epression and the value of that gives this value. Show our table below. 5. Consider the epression (a) Multipl the two sets of binomials and combine like terms in order to write this epression as an equivalent trinomial in standard form. Show our work. (b) Set up a TABLE to verif that our answer in part (a) is equivalent to the original epression. Don't hesitate to point out that it is not equivalent (which means ou either made a mistake in our algebra or in our table set up). Show our table. 6. The product of three binomials is shown below. Write this product as a polnomial in standard form. (Its highest power will be ) Set up a table for the answer ou found in #6 on the interval 5 5. Where does this epression have zeroes? Unit 1 Lesson 6 Using Tables on our Calculator Page

26 Amount Charged, A l g e b r a U n i t 1 - Foundations of Algebra LESSON 7: INTRODUCTION TO FUNCTIONS Most higher level mathematics is built upon the concept of a function. Make sure to know the following definition: DEFINITION: A function is an rule that assigns eactl one output value (-value) for each input value (value). These rules can be epressed in different was, the most common being equations, graphs, and tables of values. We call the input variable independent and output variable dependent. Eercise #1: An internet music service offers a plan whereb users pa a flat monthl fee of $5 and can then download songs for 10 cents each. (a) What are the independent and dependent variables in this scenario? Independent: Dependent: (b) Fill in the table below for a variet of independent values: Number of downloads, Amount Charged, (c) Let the number of downloads be represented b the variable and the amount charged be represented b the variable, write an equation that models as a function of (d) Based on the equation ou found in part (c), produce a graph of this function for all values of on the interval Use a calculator TABLE to generate additional coordinate pairs to the ones ou found in part (b) Number of Downloads, Unit 1 Lesson 7 Introduction to Functions Page 4

27 Relationship B Relationship A A l g e b r a U n i t 1 - Foundations of Algebra Eercise #: One of the following graphs shows a relationship where is a function of and one does not. (a) Draw the vertical line whose equation is on both graphs. (b) Give all output values for each graph at an input of. Relationship A: Relationship B: (c) Eplain which of these relationships is a function and wh. Eercise #: The graph of the function (a) State this function s -intercept. 4 1 is shown below. (b) Between what two consecutive integers does the larger - intercept lie? (c) Draw the horizontal line on this graph. (d) Using these two graphs, find all values of that solve the equation below: 41 (e) Verif that these values of are solutions b using STORE on our graphing calculator. Unit 1 Lesson 7 Introduction to Functions Page 5

28 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 7: INTRODUCTION TO FUNCTIONS HOMEWORK 1. Determine for each of the following graphed relationships whether is a function of using the Vertical Line Test. (a) (b) (c) (d) (e) (f). What are the outputs for an input of 5 given functions defined b the following formulas: (a) 4 (b) 50 (c) Unit 1 Lesson 7 Introduction to Functions Page 6

29 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.). Evan is walking home from the museum. He starts 8 blocks from home and walks blocks each minute. Evan s distance from home is a function of the number of minutes he has been walking. (a) Which variable is independent and which variable is dependent in this scenario? (b) Fill in the table below for a variet of time values. Time, t, in minutes Distance from home, D, in blocks (c) Determine an equation relating the distance, D, that Evan is from home as a function of the number of minute, t, that he has been walking. (d) Determine the number of minutes, t, that it takes for Evan to reach home. 4. In one of the following tables, the variable is a function of the variable. Eplain which relationship is a function and wh the other is not. Relationship #1 Relationship # Unit 1 Lesson 7 Introduction to Functions Page 7

30 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 8: FUNCTION NOTATION Functions are basic tools that convert inputs, values of the independent variable, to outputs, values of the dependent variable. We will use function notation all ear. The first eercise shows this notation in formulas. Eercise #1: Evaluate each of the following given the function definitions and input values. (a) f 5 (b) g 4 (c) h f g h f g 0 h Do not confuse function notation with multiplication. Function notation is summarized below. FUNCTION NOTATION Output Rule Input Function rules commonl come in one of three forms: (1) equations (as in Eercise #1), () graphs, and () tables. The net few eercises shows function notation in these three forms. Eercise #: Boiling water at 1 degrees Fahrenheit is left in a room that is at 65 degrees Fahrenheit and begins to cool. Temperature readings are taken each hour and are given in the table below. In this scenario, the temperature, T, is a function of the number of hours, h. h (hours) T h F (a) Evaluate T and T 6. (b) For what value of h is 76 T h? (c) Between what two consecutive hours will Th 100? Unit 1 Lesson 8 Function Notation Page 8

31 A l g e b r a U n i t 1 - Foundations of Algebra Eercise #: The function f is defined b the graph shown below. Use this graph to answer the following questions (a) Evaluate f 1, f 1, and f 5. (b) Evaluate f 0. What special feature on a graph does f 0 alwas correspond to? (c) What values of solve the equation f 0. What special features on a graph does the set of -values that solve f 0 correspond to? (d) Between what two consecutive integers does the largest solution to f lie? Eercise #4: For a function graph of g? g it is known that g 7. Which of the following points must lie on the (1) 7, () 0, 7 (), 7 (4), 0 Eercise #5: Phsics students drop a ball from the top of a 50 foot high building and model its height as a function of time with the equation ht 50 16t. Using TABLES on our calculator, determine, to the nearest tenth of a second, when the ball hits the ground. Provide tabular outputs to support our answer. Unit 1 Lesson 8 Function Notation Page 9

32 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 8: FUNCTION NOTATION HOMEWORK 1. Without using our calculator, evaluate each of the following given the function definitions and input values. (a) f 7 (b) g (c) h 5 f 4 g h 41 f g h 14. Using STORE on our calculator, evaluate each of the following more comple functions. (a) f (b) g 5 (c) h 01. f 5 g 4 h f 0 g h 0. Based on the graph of the function g (a) Evaluate g, g 0, g and g 7. shown below, answer the following questions. (b) What values of solve the equation g 0 (c) Graph the horizontal line on the grid above and label. (d) How man values of solve the equation g? Unit 1 Lesson 8 Function Notation Page 0

33 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.) 4. Ian invested $500 in an investment vehicle that is guaranteed to earn 4% interest compounded earl. The amount of mone, A, in his account as a function of the number of ears, t, since creating the account is given b the equation At t. (a) Evaluate A0 and A 10. (b) What do the two values that ou found in part (a) represent? (c) Using tables on our calculator, determine, to the nearest whole ear, the value of t that solves the equation At Justif our answer with numerical evidence. (d) What does the value of t that ou found in part (c) represent about Ian s investment? 5. A ball is shot from an air-cannon at an angle of 45 with the horizon. It travels along a path given b the 1 equation hd d d, where h represents the ball s height above the ground and d represents the 50 distance the ball has traveled horizontall. Using our calculator to generate a table of values, graph this function for all values of d on the interval 0 d 50. Look at the table to properl scale the -ais. What is the maimum height that the ball reaches? At what value of d does it reach this height? Unit 1 Lesson 8 Function Notation Page 1

34 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 9: FUNCTION COMPOSITION Functions convert the value of an input variable into the value of an output variable. This output could then be used as an input to a second function. This process is known as composition of functions, in other words, combining the action or rules of two functions. Eercise #1: A circular garden with a radius of 15 feet is to be covered with topsoil at a cost of $1.5 per square foot of garden space. (a) Determine the area of this garden to the nearest square foot. (b) Using our answer from (a), calculate the cost of covering the garden with topsoil. In this eercise, we see that the output of an area function is used as the input to a cost function. This idea can be generalized to generic functions, f and g as shown in the diagram below. Input = Output from f becomes the input to g Final output = There are two notations that are used to indicate composition of two functions. These will be introduced in the net few eercises, both with equations and graphs. f 5 and g, find values for each of the following. Eercise #: Given (a) f g1 (b) g f (c) g g0 (d) f g (e) g f (f) 1 f f Unit 1 Lesson 9 Function Composition Page

35 A l g e b r a U n i t 1 - Foundations of Algebra Eercise #: The graphs below are of the functions f and g. Evaluate each of the following questions based on these two graphs. (a) g f (b) f g1 (c) g g1 (d) g f (e) f g0 (f) 0 f f Unit 1 Lesson 9 Function Composition Page

36 A l g e b r a U n i t 1 - Foundations of Algebra 1. Given f 4 and g 7 (a) LESSON 9: FUNCTION COMPOSITION HOMEWORK evaluate: f g 0 (b) g f (c) f f (d) g f 6 (e) f g 5 (f) g g. Given h 11 and g evaluate: (a) h g 18 (b) g h 4 (c) g g 11 (d) hh 0 (e) h g 8 (f) g h 0. The graphs of h and k are shown below. Evaluate the following based on these two graphs. (a) hk (b) k h 0 (c) hh (d) k k Unit 1 Lesson 9 Function Composition Page 4

37 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.) 4. Scientists modeled the intensit of the sun, I, as a function of the number of hours since 6:00 a.m., h, using 1h h the function Ih. The then model the temperature of the soil, T, as a function of the intensit 6 using the function T I 5000I. Which of the following is closest to the temperature of the soil at :00 p.m.? (1) 54 () 67 () 84 (4) 8 5. Phsics students are studing the effect of the temperature, T, on the speed of sound, S. The find that the speed of sound in meters per second is a function of the temperature in degrees Kelvin, K, b S K 410K The degrees Kelvin is a function of the temperature in Celsius given b K C C 7.15 of sound when the temperature is 0 degrees Celsius. Round to the nearest tenth.. Find the speed 6. Consider the functions f 9 and g (a) g f 15 (b) 9. Calculate the following. g f (c) f ( g (9)) (d) What appears to alwas be true when ou compose these two functions? Unit 1 Lesson 9 Function Composition Page 5

38 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 10: THE DOMAIN AND RANGE OF A FUNCTION Because functions convert values of inputs into value of outputs, we talk about the sets that represent these inputs and outputs. The set of inputs that result in an output is called the domain of the function. The set of outputs is called the range. Eercise #1: Consider a function that has as its inputs the months of the ear, and as its outputs the number of das in each month. In this case, the number of das is a function of the month of the ear. Assume this function is restricted to non-leap ears. (a) Write, in roster form, the set that represents this function s domain. (b) Write, in roster form, the set that represents this function s range. Eercise #: State the range of the function f n n 1 if its domain is the set 1,, 5. Show the domain and range in the mapping diagram below. Domain of f Range of f Eercise #: The function g is completel defined b the graph shown below. Answer the following questions based on this graph. (a) Determine the minimum and maimum -values represented on this graph. (b) Determine the minimum and maimum -values represented on this graph. (c) State the domain and range of this function using set builder notation. Unit 1 Lesson 10 Domain & Range of Functions Page 6

39 A l g e b r a U n i t 1 - Foundations of Algebra Some functions, defined with graphs or equations, have domains and ranges that stretch out infinitel. Consider the following eercise in which a standard parabola is graphed. Eercise #4: The function f 1is graphed on the grid below. Which of the following represent its domain and range written in interval notation? (1) Domain:, 4 Range: 4, 6 () Domain:, 4 Range: 4, () Domain:, Range: 4, (4) Domain:, 4 Range: 4, 6 For most functions defined b an algebraic formula, the domain consists of the set of all real numbers, given the concise smbol R. Sometimes, though, there are restrictions placed on the domain of a function b the structure of its formula. Two basic restrictions will be illustrated in the net few eercises. Eercise #5: The function f (a) Evaluate f 1 and f 6 from the table. 1 has outputs given b the following calculator table. 4 f (b) Wh does the calculator give an ERROR at 4? -7 4 Error (c) Are there an values ecept 4 that are not in the domain of f? Eplain Eercise #6: Which of the following values of would not be in the domain of the function 4? Eplain our answer. (1) 0 () () 5 (4) 8 Unit 1 Lesson 10 Domain & Range of Functions Page 7

40 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 10: THE DOMAIN AND RANGE OF A FUNCTION HOMEWORK 1. A function is given b the set of ordered pairs in roster form., 5, 4, 9, 6,1, 8,17. Write its domain and range Domain: Range:. The function h 5 maps the domain given b the set, 1, 0,1, represents the range of h? (1) 0, 6,10,1 () 5, 6, 9 () 5, 6, 7 (4) 1, 4, 5, 6, 9. Which of the following sets. Which of the following values of would not be in the domain of the function defined b f (1) () () (4) 4. Determine an values of that do not lie in the domain of the function f response.?. Justif our Which of the following values of does lie in the domain of the function defined b g 7 (1) 0 () () (4) 5? 6. Which of the following would represent the domain of the function 6? (1) : () : () : (4) : Unit 1 Lesson 10 Domain & Range of Functions Page 8

41 A l g e b r a U n i t 1 - Foundations of Algebra 7. The function f HOMEWORK (cont.) is completel defined b the graph shown below. (a) Evaluate f 4, f, and f 6. (b) Draw a rectangle that circumscribes (just surrounds) the graph. (c) State the domain and range of this function using interval notation. Domain: Range: 8. Which of the following represents the range of the quadratic function shown in the graph below? (1) 4, (),4 (),4 (4) 4, 9. A child starts a pigg bank with $. Each da, the child receives 5 cents at the end of the da and puts it in the bank. If A represents the amount of mone and d stands for the number of das then Ad 0.5 gives the amount of mone in the bank as a function of das (think about this formula). d (a) Evaluate. (b) For what value of d will. (c) Eplain wh the domain does not contain the value. (d) Eplain wh the range does not include the value. Unit 1 Lesson 10 Domain & Range of Functions Page 9

42 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 11: ONE-TO-ONE FUNCTIONS A special categor of functions is called one-to-one. The following eercise shows the difference between a function that is one-to-one and one that is not. f and g. Eercise #1: Consider the two functions given b the equations (a) Map the domain, 0, using each function. Fill in the range and show the mapping arrows. Domain of f Range of f Domain of g Range of g (b) What is different between these two functions in terms of how the elements of this domain get mapped to the elements of the range? ONE-TO-ONE FUNCTIONS A function is called one-to-one if implies that. In other words, different inputs alwas give different outputs. Eercise #: Of the four tables below, one represents a relationship where is a one-to-one function of. Determine which it is and eplain wh the others are not. (1) () () (4) Unit 1 Lesson 11 One to One Functions Page 40

43 A l g e b r a U n i t 1 - Foundations of Algebra Eercise #: Consider the following four graphs which show a relationship between the variables and. (1) () (a) Circle the two graphs above that are functions. Eplain how ou know the are functions. () (4) (b) Of the two graphs ou circled, which is oneto-one? Eplain how ou can tell from its graph. THE HORIZONTAL LINE TEST If an horizontal line passes through the graph of a function at most one time, then that function is one-to-one. This test works because horizontal lines represent constant -values. If a horizontal line intersects a graph more than once, an output has been repeated, and the function is not one-to-one. Eercise #4: Which of the following represents the graph of a one-to-one function? (1) () () (4) Eercise #5: The distance that a number,, lies from the number 5 on a one-dimensional number line is given b the function D 5 D is not a one-to-one function.. Show b eample that Unit 1 Lesson 11 One to One Functions Page 41

44 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 11: ONE-TO-ONE FUNCTIONS HOMEWORK 1. Which of the following graphs illustrates a one-to-one relationship? (1) () () (4). Which of the following graphs does not represent that of a one-to-one function? (1) () () (4). In which of the following graphs is each input not paired with a unique output? (1) () () (4) 4. In which of the following formulas is the variable a one-to-one function of the variable? (Hint tr generating some values either in our head or using TABLES or graphs on our calculator.) (1) () () (4) 5 Unit 1 Lesson 11 One to One Functions Page 4

45 A l g e b r a U n i t 1 - Foundations of Algebra HOMEWORK (cont.) 5. Which of the following tables illustrates a relationship in which is a one-to-one function of? (1) () () (4) A recent newspaper gave temperature data for various das of the week in table format. In which of the tables below is the reported temperature a one-to-one function of the da of the week? (1) () () (4) Mon 75 Tue 68 Wed 65 Thu 74 Mon 75 Tue 7 Wed 68 Thu 7 Mon 58 Tue 5 Mon 81 Tue 76 Mon 56 Tue 58 Mon 85 Tue Phsics students drop a basketball from 5 feet above the ground and its height is measured each tenth of a second until it stops bouncing. The height of the basketball, h, is clearl a function of the time, t, since it was dropped. (a) Sketch the general graph of what ou believe this function would look like. (b) Is the height of the ball a one-to-one function of time? Eplain our answer. 5 h (ft) t (sec) 8. Consider the function f ( ) round( ), which rounds the input,, to the nearest integer. Is this function one-to-one? Eplain or justif our answer. Unit 1 Lesson 11 One to One Functions Page 4

46 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 1: KEY FEATURES OF FUNCTIONS The graphs of functions have man ke features whose terminolog we will use all ear. Eercise #1: The function f is shown graphed to the right. Answer the following questions based on this graph. (a) State the -intercept of the function. (b) State the -intercepts of the function. What is the alternative name that we give the -intercepts? (c) Over the interval 1 How can ou tell? is f increasing or decreasing? f (d) Give the interval over which f 0. What is a quick wa of seeing this visuall? (e) State all the -coordinates of the relative maimums and relative minimums. Label each. (f) What are the absolute maimum and minimum values of the function? Where do the occur? (g) State the domain and range of interval notation. f using (h) If a second function g is defined b the formula g 1 f, then what is the -intercept of g? Unit 1 Lesson 1 Ke Features of Functions Page 44

47 A l g e b r a U n i t 1 - Foundations of Algebra Eercise #: Consider the function g 1 8 the domain 4 7. defined over (a) Sketch a graph of the function to the right. (b) State the domain interval over which this function is decreasing. (c) State zeroes of the function on this interval. (d) State the interval over which g 0 g 0 b using the algebraic definition of the function. What point does this correspond to on the graph? (e) Evaluate (f) Are there an relative maimums or minimums on the graph? If so, which and what are their coordinates? You need to be able to think about functions in all of their forms, including equations, graphs, and tables. Tables can be quick to use, but sometimes hard to understand. Eercise #: A continuous function f has a domain of 6 1 with selected values shown below. The function has eactl two zeroes and has eactl two turning points, one at, 4 and one at 9, f (a) State the interval over which f 0. (b) State the interval over which increasing. f is Unit 1 Lesson 1 Ke Features of Functions Page 45

48 A l g e b r a U n i t 1 - Foundations of Algebra LESSON 1: KEY FEATURES OF FUNCTIONS HOMEWORK 1. The function f is shown to the right. Answer the following questions based on its graph. (a) Evaluate each of the following based on the graph: (i) f 4 (ii) f (b) State the zeroes of f. (c) Over which of the following intervals is increasing? f alwas (1) 7 () 5 5 () 5 (4) 5 (d) State the coordinates of the relative maimum and the relative minimum of this function. Relative Maimum: Relative Minimum: g is defined using the rule g f 5. Evaluate g 0. What (f) A second function does this correspond to on the graph of g? (e) Over which of the following intervals is f 0? (1) 7 () 5 () 7 (4) 5 (g) A third function h is defined b the formula h. What is the value of g h? Show how ou arrived at our answer. Unit 1 Lesson 1 Ke Features of Functions Page 46

49 A l g e b r a U n i t 1 - Foundations of Algebra. For the function g 9 1 HOMEWORK (cont.) do the following. (a) Sketch the graph of g on the aes provided. (b) State the zeroes of g. (c) Over what interval is g decreasing? (d) Over what interval is g 0? (e) State the range of g.. Draw a graph of f characteristics. that matches the following Increasing on: 8 4 and 1 5 Decreasing on: 4 1 f 8 5 and zeroes at 6,, and Absolute maimum of 7 and absolute minimum of 5 4. A continuous function has a domain of 7 10 and has selected values shown in the table below. The function has eactl two zeroes and a relative maimum at 4,1 and a relative minimum at 5, 6. (a) State the interval on which f f is decreasing. (b) State the interval over which f 0. Unit 1 Lesson 1 Ke Features of Functions Page 47

50 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra U n i t - Linear Functions, Equations, and their Algebra LESSON 1: AVERAGE RATE OF CHANGE We begin our linear unit b studing the rate of change of functions. When we model using functions, we are ver often interested in the rate that the output is changing compared to the rate of the input. Eercise #1: The function f is shown graphed to the right. (a) Evaluate each of the following based on the graph: (i) f 0 (ii) f 4 (iii) f 7 (iv) f 1 (b) Find the change in the function, f, over each of the following domain intervals. Find this both b subtraction and show this on the graph. (i) 0 4 (ii) 4 7 (iii) 7 1 (c) Wh can't ou simpl compare the changes in f from part (b) to determine over which interval the function changing the fastest? (d) Calculate the average rate of change for the function over each of the intervals and determine which interval has the greatest rate. (i) 0 4 (ii) 4 7 (iii) 7 1 (e) Using a straightedge, draw in the lines whose slopes ou found in part (d) b connecting the points shown on the graph. The average rate of change gives a measurement of what propert of the line? Unit Lesson 1 Average Rate of Change Page 48

51 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra The average rate of change is a ver important mathematical concept because it gives us a wa to quantif how fast a function changes, on average, over a certain domain interval. We used its formula in the last eercise: AVERAGE RATE OF CHANGE For a function over the domain interval, the function's average rate of change is calculated b: Eercise #: Consider the two functions f 5 7 and g 1. (a) Calculate the average rate of change for both functions over the following intervals. Do our work carefull and show the calculations that lead to our answers. (i) (ii) 1 5 (b) The average rate of change for f was the same for both (i) and (ii) but was not the same for g. Wh is that? Eercise #: The table below represents a linear function. Fill in the missing entries Unit Lesson 1 Average Rate of Change Page 49

52 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON 1: AVERAGE RATE OF CHANGE HOMEWORK 1. For the function g given in the table below, calculate the average rate of change for each of the following intervals g (a) 1 (b) 1 6 (c) 9 (d) Eplain how ou can tell from the answers in (a) through (c) that this is not a table that represents a linear function. f. Consider the simple quadratic function the following intervals:. Calculate the average rate of change of this function over (a) 0 (b) 4 (c) 4 6 (d) Clearl the average rate of change is getting larger at gets larger. How is this reflected in the graph of f shown sketched to the right? Unit Lesson 1 Average Rate of Change Page 50

53 Distance (feet) A l g e b r a U n i t - Linear Functions, Equations, and their Algebra HOMEWORK (cont.) the function g 16. Which has a greater average rate of change over the interval 4 f function. Provide justification for our answer. or the 4. An object travels such that its distance, d, awa from its starting point is shown as a function of time, t, in seconds, in the graph below. (a) What is the average rate of change of d over the interval 5t 7? Include proper units in our answer. (b) The average rate of change of distance over time (what ou found in part (a)) is known as the average speed of an object. Is the average speed of this object greater on the interval 0t 5 or 11t 14? Justif. Time (seconds) 5. What makes the average rate of change of a linear function different from that of an other function? What is the special name that we give to the average rate of change of a linear function? Unit Lesson 1 Average Rate of Change Page 51

54 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON : FORMS OF A LINE Linear functions come in a variet of forms. The two shown below have been introduced in Common Core Algebra I and Common Core Geometr. TWO COMMON FORMS OF A LINE Slope-Intercept: where m is the slope (or average rate of change) of the line and Point-Slope: represents one point on the line. Eercise #1: Consider the linear function f 5. (a) Determine the -intercept of this function b evaluating. (b) Find its average rate of change over the interval. Eercise #: Consider a line whose slope is 5 and which passes through the point, 8. (a) Write the equation of this line in point-slope form,. (b) Write the equation of this line in slope-intercept form Eercise #: Which of the following represents an equation for the line that is parallel to passes through the point 6, 8? (1) 8 6 () 8 6 () 8 6 (4) and which Unit Lesson Forms of a Line Page 5

55 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra Eercise #4: A line passes through the points 5, and 0, 4. (a) Determine the slope of this line in simplest rational form. (b) Write an equation of this line in point-slope form. (c) Write an equation for this line in slopeintercept form. (d) For what -value will this line pass through a -value of 1? Eercise #5: The graph of a linear function is shown below. (a) Write the equation of this line in m b form. (b) What must be the slope of a line perpendicular to the one shown? (c) Draw a line perpendicular to the one shown that passes through the point 1,. (d) Write the equation of the line ou just drew in pointslope form. (e) Does the line that ou drew contain the point 0, 15? Justif. Unit Lesson Forms of a Line Page 5

56 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra 1. Which of the following lines is perpendicular to (1) () 5 4 () 4 (4) 5 LESSON : FORMS OF A LINE HOMEWORK Which of the following lines passes through the point 4, 8? (1) 8 4 () 8 4 () 8 4 (4) and has a -intercept of 4?. Which of the following equations could describe the graph of the linear function shown below? (1) () 4 () 4 (4) For a line whose slope is and which passes through the point 5, : (a) Write the equation of this line in point-slope m. form, 1 1 (b) Write the equation of this line in slopeintercept form, m b. 5. For a line whose slope is 0.8 and which passes through the point,1 : (a) Write the equation of this line in point-slope m. form, 1 1 (b) Write the equation of this line in slopeintercept form, m b. Unit Lesson Forms of a Line Page 54

57 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra HOMEWORK (cont.) 6. The two points, 6 and 6, 0 are plotted on the grid below. (a) Find an equation, in m b form, for the line passing through these two points. Use of the grid is optional. (b) Does the point 0, 16 lie on this line? Justif. 7. A linear function is graphed below along with the point,1. (a) Draw a line parallel to the one shown that passes through the point,1. (b) Write an equation for the line ou just drew in point-slope form. (c) Between what two consecutive integers does the -intercept of the line ou drew fall? (d) Determine the eact value of the -intercept of the line ou drew. Unit Lesson Forms of a Line Page 55

58 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON : INVERSE FUNCTIONS The idea of inverses, or opposites, is ver important in math. The word inverse is used in man different contets, including the additive inverse and multiplicative inverse of a number. The actions of certain functions can be reversed. The rule of a function s reversal can also be a function. 7 7 Eercise #1: Consider the two linear functions given b the formulas f and g. 7 (a) Calculate f 5 and g11. (b) Calculate f 0 and g. (c) Calculate f. g1 (d) Calculate f g 5. (e) Without calculation, determine the value of f g. The two functions seen in Eercise #1 are inverses because the literall undo one another. The general idea of inverses, f and g, is shown below in the mapping diagram. Domain of f Range of f a b Range of g Domain of g Eercise #: If the point, 5 lies on the graph of f, then which of the following points must lie on the graph of its inverse? 5, (1), 5 () () 5, (4) 1 1, 5 Unit Lesson Inverse Functions Page 56

59 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra Inverse functions have their own special notation, as follows: INVERSE FUNCTION NOTATION If a function has an inverse that is also a function we represent it as. Eercise #: The linear function questions. f is shown graphed below. Use its graph to answer the following f (a) Evaluate 1 1 and f 4. (b) Determine the -intercept of f. 1 (c) On the same set of aes, draw a graph of f. 1 (d) Write the equation of f 1 ( ) Eercise #4: A table of values for the simple quadratic function f() (a) Graph the inverse b switching the ordered pairs. f 1 ( ) f ( ) is given below along with its graph. (b) What do ou notice about the graph of this function s inverse? EXISTENCE OF INVERSE FUNCTIONS LESSON 6: INVERSE FUNCTIONS A function will have an inverse that is also a function if and onl if it is one-to-one. A quick wa to know if a function has an inverse that is also a function HOMEWORK is to appl the Horizontal Line Test. Unit Lesson Inverse Functions Page 57

60 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON : INVERSE FUNCTIONS HOMEWORK 1. If the point 7, 5 lies on the graph of f, which of the following points must lie on the graph of its inverse? (1) 5, 7 () 7, (), (4) , The function f has an inverse function f. If f a b then which of the following must be true? (1) f b a () 1 1 f b a () f (4) a b f b a 1. The graph of the function g is shown below. The value of g 1 is (1).5 () 0.4 () 4 (4) 1 4. Which of the following functions would have an inverse that is also a function? (1) () () (4) 5. For a one-to-one function it is known that 0 6 and f 8 0. Which of the following must be true about the graph of this function s inverse? (1) its -intercept = 6 () its -intercept = 6 () its -intercept = 8 (4) its -intercept = 8 f Unit Lesson Inverse Functions Page 58

61 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra HOMEWORK (cont.) 6. The function h is entirel defined b the graph shown: (a) Sketch a graph of h 1 (b) Write the domain and range of using interval notation.. Create a table of values if needed. 1 h and h h Domain: Range: h 1 Domain: Range: 7. The function A r r is a one-to-one function that uses a circle s radius as an input and gives the circle s area as its output. Selected values of this function are shown in the table below. r Ar (a) Determine the values of from using the table. A 9 and A (b) Determine the values of A A and (c) The original function A r What are the inputs and outputs of the inverse function? converted an input, the circle s radius, to an output, the circle s area. Input: Output: 8. The domain and range of a one-to-one function, f, are given below in set-builder notation. Give the domain and range of this function s inverse also in set-builder notation. f 1 f Domain: 5 Range: Domain: Range: Unit Lesson Inverse Functions Page 59

62 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON 4: INVERSES OF LINEAR FUNCTIONS Recall that one-to-one functions have inverses that are also functions. Ecept for horizontal lines, all linear functions are one-to-one and thus have inverses that are also functions. In this lesson we will investigate these inverses and how to find their equations. Eercise #1: On the grid below the linear function 4 is graphed along with the line. (a) How can ou quickl tell that 4 is a one-to-one function? (b) Graph the inverse of 4 on the same grid. Recall that this is easil done b switching the and coordinates of the original line. (c) What can be said about the graphs of 4 and its inverse with respect to the line? (d) Find the equation of the inverse in m b form. As we can see from part (e) in Eercise #1, inverses of linear functions include the inverse operations of the original function but in reverse order. The simple method of finding the equation of an inverse is to simpl switch the and variables in the original equation and solve for. Eercise #: Which of the following represents the equation of the inverse of 5 0? (1) () 1 0 () (4) Unit Lesson 4 Inverses of Linear Functions Page 60

63 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra Although this is a simple procedure, common errors are often made when solving for. Be careful with each algebraic step. Eercise #: Which of the following represents the inverse of the linear function (1) () 8 () 1 (4) 8 1 8? Eercise #4: What is the -intercept of the inverse of (1) 15 () 9 9? 5 () 1 (4) 9 5 Sometimes we are asked to work with linear functions in their point-slope form. The method of finding the inverse and plotting it, though, do not change just because the linear equation is written in a different form. Eercise #5: Which of the following would be an equation for the inverse of (1) 6 () () 6 (4) 4 6? Eercise #6: Which of the following points lies on the graph of the inverse of 8 5 choice. (1) 8, () 10, 40 () 8, (4), 8? Eplain our Eercise #7: Which of the following linear functions would not have an inverse that is also a function? Eplain how ou made our choice. (1) () () (4) 5 1 Unit Lesson 4 Inverses of Linear Functions Page 61

64 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON 4: INVERSES OF LINEAR FUNCTIONS HOMEWORK 1. The graph of a function and its inverse are alwas smmetric across which of the following lines? (1) 0 () () 0 (4) 1. Which of the following represents the inverse of the linear function 4? (1) () 1 8 () 1 8 (4) If the -intercept of a linear function is 8, then we know which of the following about its inverse? (1) Its -intercept is 8. () Its -intercept is 1 8. () Its -intercept is 8. (4) Its -intercept is If both were plotted, which of the following linear functions would be parallel to its inverse? Eplain our thinking. (1) () 5 1 () 4 (4) 6 5. Which of the following represents the equation of the inverse of (1) () 4 4 () 18 (4) Which of the following points lies on the inverse of 4 1 (1), 1 () 1,1 () 1, (4),1? 4 4? Unit Lesson 4 Inverses of Linear Functions Page 6

65 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra HOMEWORK (cont.) 7. A linear function is graphed below. Answer the following questions based on this graph. (a) Write the equation of this linear function in m b form. (b) Sketch a graph of the inverse of this function on the same grid. (c) Write the equation of the inverse in m b form. (d) What is the intersection point of this line with its inverse? 8. A car traveling at a constant speed of 58 miles per hour has a distance of -miles from Poughkeepsie, NY, given b the equation 58 4, where represents the time in hours that the car has been traveling. (a) Find the equation of the inverse of this linear function in = m = b form. (b) Evaluate the function ou found in part (a) for an input of. (c) Give a phsical interpretation of the answer ou found in part (b). Consider what the input and output of the inverse represent in order to answer this question. 9. Given the general linear function m b, find an equation for its inverse in terms of m and b. Unit Lesson 4 Inverses of Linear Functions Page 6

66 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON 5: SYSTEMS OF LINEAR EQUATIONS Sstems of equations, or more than one equation, arise frequentl in mathematics. To solve a sstem means to find all values that simultaneousl make all equations true. Of special importance are sstems of linear equations. Eercise #1: Solve the following two-b-two () sstem of equations. Epress answer in (,) form. Check. 9 7 In this lesson, we will etend to linear sstems of equations and unknowns. Steps to solving a sstem of equations: 1. Label the equations A, B, and C. Choose a variable to eliminate (, or z).. Add/subtract sets of equations producing equations: label them D and E. 4. Solve equations D and E as a sstem to get 1 variable answer. 5. Substitute into D or E to get nd variable answer. 6. Substitute into A, B or C to get rd. 7. Check in original equations. Eercise #: Consider the sstem of linear equations shown below. Each equation is labeled A, B and C: (A) + + z = 15 (B) 6 z = 5 (C) z = 14 Unit Lesson 5 Sstems of Linear Equations Page 64

67 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra Eercise #: Solve the sstem of equations shown below. Epress answer in (,,z) form. Check answer. 4 z 6 4 z 8 5 7z 19 (a) Which variable will be easiest to eliminate? Wh? Eercise #4: Solve the sstem of equations shown below. Epress answer in (,,z) form. Check answer. 4 z 5 z 7 4z 7 Unit Lesson 5 Sstems of Linear Equations Page 65

68 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra LESSON 5: SYSTEMS OF LINEAR EQUATIONS HOMEWORK 1. Show that 10, 4, and z 7 (10,4,7) is a solution to the sstem below without solving the sstem. z 5 4 5z 1 8z. Solve the following sstem of equations. Show all work. Epress answer in (,,z) form. Check answer. 4 z 1 z 1 5z 70 Unit Lesson 5 Sstems of Linear Equations Page 66

69 A l g e b r a U n i t - Linear Functions, Equations, and their Algebra HOMEWORK (cont.). Solve the following sstem of equations. Show all work. Epress answer in (,,z) form. Check answer. 5 z 5 4z 1 z 4. Solve the following sstem of equations. Show all work. Epress answer in (,,z) form. Check answer. z 4 5 z z 50 Unit Lesson 5 Sstems of Linear Equations Page 67

70 A l g e b r a U n i t - Eponential Functions U n i t - Eponential Functions LESSON 1: EXPONENTIAL PROPERTIES Rules of Eponents and Eponents that are Not Positive Let a and b be positive integers and and real numbers: a b a. Multiplication Rule e. a 4 a = a b. Division Rule b 5 e. 4 a b c. Power Rule ( ) e. (a 4 ) = d. Power of a Product *Note ( z) e. Power of a Quotient a a ( z) = e. (abc) = a a = e. = f. Zero Eponent 0 = e. 9,999 0 = g. Zero eponent w/coefficient a 0 = e 9,999 0 = a h. Negative Eponent e. - = i. Fraction raised to a negative eponent a = e. = 4 a j. Fraction w/neg. eponent in numerator: e. b 7 b 7 k. Fraction w/neg. eponent in denominator: a e. Unit Lesson 1 Eponential Properties Page 68

71 A l g e b r a U n i t - Eponential Functions Eamples: Simplif 4 1. ( ) 5. 4 ( ) ( ) ( ) k (4-4 )(4 - ) Unit Lesson 1 Eponential Properties Page 69

72 A l g e b r a U n i t - Eponential Functions LESSON 1: EXPONENTIAL PROPERTIES HOMEWORK 1. Epress each of the following epressions in "epanded" form, i.e., do all of the multiplication and/or division possible and combine as man eponents as possible. (a) (b) 1 (c) 5 7 (d) (e) 9 (f) (g) 10 (h) (i) 8 (j) (k) 4 (l) 4 (m) (n) (o) (p) (q) (r) (s) (t) (u) Unit Lesson 1 Eponential Properties Page 70

73 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) Unit Lesson 1 Eponential Properties Page 71

74 A l g e b r a U n i t - Eponential Functions LESSON : Eponential Properties Practice Multiplication: More eamples: Write the epression without a denominator. 4a b 4a b 1. =. 4 4 a a = Write the epression using onl positive eponents. 5 z. = = 7 4a bc 5. 7 a b c = = Describe the mistake: 7. (-)²(-)³ = Unit Lesson Eponential Properties Practice Page 7

75 A l g e b r a U n i t - Eponential Functions Evaluate each function for the given value. Don t use a calculator. 1. f() = 1 f() =. f() = + 4 ; f(1) =. f() = ( + 6 ) ; f( ) = 0 4. f() = 1 ; f() = Simplif: z a b 5 1 c Unit Lesson Eponential Properties Practice Page 7

76 A l g e b r a U n i t - Eponential Functions LESSON : EXPONENTIAL PROPERTIES PRACTICE HOMEWORK If f 5 4 then f a (1) 1a 5 1 () 5 4a 4 () 5 a (4) 1a Which of the following is equivalent to 5 6 lead to our final answer (1) 9 () 14 for all 0? Show the manipulations that () (4) 4 Unit Lesson Eponential Properties Practice Page 74

77 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) In 9-50, write each epression with onl positive eponents. Epress the answer in simplest form. In 67-7 write each quotient as a product without a denominator. Unit Lesson Eponential Properties Practice Page 75

78 A l g e b r a U n i t - Eponential Functions LESSON : RATIONAL Eponents Toda we will introduce rational ( ) eponents and etend our eponential knowledge that much further. If n is a positive integer then m n = m n n m or 1 = 1 = 1 4 = = = 5 = 5 = a 4 = To evaluate an epression containing a fractional eponent: 1. Rewrite the epression in radical form.. Evaluate the root.. Evaluate the power. Practice: Evaluate and/or simplif without a calculator =. (7) = 1. (16) 1 = = 5. (-15) 1 1 = = 7. ( 7) = 8. (16) 4 = 9. ( 8) = 10. (7) = Unit Lesson Rational Eponents Page 76

79 A l g e b r a U n i t - Eponential Functions = (² ³) Rewrite using rational eponents. Then simplif our answer ab 7 z 4 Evaluate each function for the given value. 1. f() = + 1 f(9) =. f() = 4 4 ; f(8) = Note: If n (the inde of the radical) is an even number, the base,, cannot be negative. However, if n is an odd number, the base,, can be negative. That s because if n is even, an imaginar number would result. For eample: 1 ( 8) = 8 = (a real number) However, 1 ( 16) = 16 = 4i (an imaginar number) Unit Lesson Rational Eponents Page 77

80 A l g e b r a U n i t - Eponential Functions LESSON : RATIONAL EXPONENTS HOMEWORK 1. Rewrite the following as equivalent roots and then evaluate as man as possible without our calculator. (a) 1 6 (b) 1 7 (c) 1 5 (d) (e) (f) 1 49 (g) (h) 1 4. Evaluate each of the following b considering the root and power indicated b the eponent. Do as man as possible without our calculator. (a) 8 (b) 4 (c) 4 16 (d) (e) 5 4 (f) 7 18 (g) 4 65 (h) 5 4. Given the function (1) 40 () 4 () 0 (4) 0 f 5 4, which of the following represents its -intercept? Unit Lesson Rational Eponents Page 78

81 4. Which of the following is equivalent to (1) A l g e b r a U n i t - Eponential Functions 1 () 1 () (4) 1 HOMEWORK (cont.) 1? 5. Written without fractional or negative eponents, (1) () 1 () (4) 1 1 is equal to 6. Which of the following is not equivalent to (1) 4096 () 64 16? () 8 (4) Marlene claims that the square root of a cube root is a sith root? Is she correct? To start, tr rewriting the epression below in terms of fractional eponents. Then appl the Product Propert of Eponents. a 1 8. f() = () + 0 ; f(9) = Unit Lesson Rational Eponents Page 79

82 A l g e b r a U n i t - Eponential Functions LESSON 4: MORE PRACTICE WITH RATIONAL Eponents All rules of eponents appl to fractional (rational) eponents. These rules justif man standard manipulations with square roots (and other radicals). For eample, simplifing roots: It s eas to simplif these: 9 49 z However, if the radicand is not a perfect square, there is another step to take. We onl consider a square root "simplified" when all of its perfect square factors have had their square roots evaluated. To Simplif a Radical: 1. Epress the radicand as the product of factors - a) the largest perfect (square or cube or n th ) factors, followed b: b) the non-perfect (square or cube or n th ) factors. Find the root of the perfect factors from (a). This is the coefficient of the radical answer. The (b) part is the radicand of the radical answer. Perfect Squares - 4, 9, 16, 5, 6, 49, 64, 81, 100, 11, 144, 169, 196, 5, ,... Perfect Cubes - 8, 7, 64, 15, 16, 4, 51, 79, 1000, Other powers to know - 4 = 4 = 5 = 6 = Simplif each of the following square roots. Show the manipulations that lead to our answers =. 700 =. 4 7 = 4. 5 = = = = = a b Unit Lesson 4 More practice with Rational Eponents Page 80

83 A l g e b r a U n i t - Eponential Functions We can etend the simplifing process to include cube roots and higher-order roots. 1. Simplif each of the following higher order roots: (a) 16 (b) 108 (c) 50 (d) 18 8 (e) 4 16 (f) (g) (h) Simplif: (a) 18 7 (b) (c) (d) Unit Lesson 4 More practice with Rational Eponents Page 81

84 A l g e b r a U n i t - Eponential Functions LESSON 4: MORE PRACTICE WITH RATIONAL EXPONENTS HOMEWORK 1. Which of the following is not equivalent to 9? (1) () 9 () 9 (4) 4. The radical epression 5 50 can be rewritten equivalentl as (1) 5 () 5 () 5 (4) If the function of ab? 1 was placed in the form b a then which of the following is the value (1) 6 () 6 () 4 (4) 4 4. Rewrite each of the following epressions without roots b using fractional eponents. (a) (b) (c) 7 (d) 5 (e) 11 1 (f) 4 1 (g) (h) Rewrite each of the following without the use of fractional or negative eponents, using radicals. (a) 1 6 (b) 1 10 (c) 1 (d) 1 5 Unit Lesson 4 More practice with Rational Eponents Page 8

85 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) 6. Simplif each of the following square roots that contain variables in the radicand. (a) 9 8 (b) Epress each of the following roots in simplest radical form. (a) 16 8 (b) Mikala was tring to rewrite the epression 1 5 in an equivalent form that is more convenient to use. She incorrectl rewrote it as 5. Eplain Mikala's error. 9. If the epression 1 was placed in a form, then which of the following would be the value of a? (1) () 1 () (4) 1 Unit Lesson 4 More practice with Rational Eponents Page 8

86 A l g e b r a U n i t - Eponential Functions LESSON 5: MORE EXPONENT PRACTICE It is important to be able to manipulate epressions involving eponents, whether those eponents are positive, negative, or fractional. The basic laws of eponents are shown below. The appl regardless of the nature of the eponent (i.e. positive, negative, or fractional). EXPONENT LAWS 1. a b = 5. a = and. a b = 6. ( )a =. ( a ) b = 7. 0 = 4. m n Τ = 8. an b n = (For integers m and n) 1 a = Although these problems can be challenging, the ke will be to carefull appl these eponent laws sstematicall. Eample 1: Simplif each of the following epressions. Leave no negative eponents in our answers. (a) 5 4 (b) (c) 4 4 (d) 6 4 In the last eample, all powers were integers. The net eample introduces fractional powers. Remember that the still follow the eponent rules above. If needed, use our calculator to help add and subtract the powers. Eample : Simplif each of the following epressions. Write each without the use of negative eponents (a) (b) 5 (c) Unit Lesson 5 More Eponent practice Page 84

87 A l g e b r a U n i t - Eponential Functions Don t forget that fractional eponents have an equivalent interpretation as radicals. You should be able to move from one representation to another. Eample : Rewrite each epression below in both its simplest form and using radical epressions. (a) 5 (b) (c) (d) 8 (f) 1 (e) Eample 4: Which of the following is equivalent to 8 7? (1) 7 8 () 7 () 7 (4) 7 8 Eample 5: The epression 1 4 is the same as (1) 1 1 () 1 4 () 1 (4) 1 1 Unit Lesson 5 More Eponent practice Page 85

88 A l g e b r a U n i t - Eponential Functions LESSON 5: MORE EXPONENT PRACTICE HOMEWORK 1. Rewrite each of the following epressions in simplest form and without negative eponents (a) (b) (c) (d) Which of the following represents the value of (1) 4 9 () 1 6 a b 4 when a and b () 4 81? (4) 1. Simplif each epression below so that it contains no negative eponents. Do not write the epressions using radicals (a) (b) 4 (c) Which of the following represents the epression 4 (1) () () written in simplest form? 4 (4) 4 Unit Lesson 5 More Eponent practice Page 86

89 A l g e b r a U n i t - Eponential Functions HOMEWORK, CONT. 5. Rewrite each of the following epressions using radicals. Epress our answers in simplest form. (a) 4 (b) 4 (c) 5 (d) (e) 5 (f) Which of the following is equivalent to 0? (1) 1 4 () () 4 5 (4) When written in terms of a fractional eponent the epression is (1) 7 () 1 () 5 (4) 8. Epressed as a radical epression, the fraction (1) 6 1 () is 1 () (4) 6 11 Unit Lesson 5 More Eponent practice Page 87

90 A l g e b r a U n i t - Eponential Functions LESSON 6: EXPONENTIAL FUNCTION BASICS This lesson reviews man of the basic components of eponential graphs and behavior. Eponential functions, those whose eponents are variable, are etremel important in math, science, and engineering. Eercise #1: Consider the function sketch the graph on the grid provided Eercise #: Now consider the function below and sketch the graph on the aes provided. BASIC EXPONENTIAL FUNCTIONS where. Fill in the table below without using our calculator and then 1. Using our calculator to help ou, fill out the table Unit Lesson 6 Eponential Function Basics Page 88

91 A l g e b r a U n i t - Eponential Functions Eercise #: Based on the graphs and behavior ou saw in Eercises #1 and #, state the domain and range for an eponential function of the form b. Domain (input set): Range (output set): Eercise #4: Are eponential functions one-to-one? How can ou tell? What does this tell ou about their inverses? Eercise #5: Now consider the function 7. (a) Determine the -intercept of this function algebraicall. Justif our answer. (b) Does the eponential function increase or decrease? Eplain our choice. (c) Create a rough sketch of this function, labeling its - intercept Eercise #6: Consider the function (a) How does this function s graph compare to that of 1? What does adding 4 do to a function's graph? (b) Determine this graph s -intercept algebraicall. Justif our answer. (c) Create a rough sketch of this function, labeling its - intercept. Unit Lesson 6 Eponential Function Basics Page 89

92 A l g e b r a U n i t - Eponential Functions LESSON 6: EXPONENTIAL FUNCTION BASICS HOMEWORK 1. Which of the following represents an eponential function? (1) 7 () () 7 (4) f then f. If 69 withou a calculator.) 1 7 7? (Remember what we just learned about fractional eponents and do (1) 7 () 7 () 18 (4) 15. If h and g 5 7 then h g (1) 18 () 8 () 1 (4) 7 4. Which of the following equations could describe the graph shown below? (1) 1 () 1 () (4) 4 5. Which of the following equations represents the graph shown? () (1) 5 1 () 4 1 (4) Unit Lesson 6 Eponential Function Basics Page 90

93 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) 6. Sketch graphs of the equations shown below on the aes given. Label the -intercepts of each graph. (a) 18 1 (b) The Fahrenheit temperature of a cup of coffee, F, starts at a temperature of 185 F. It cools down according to the eponential function Fm m 0 (a) How do ou interpret the statement that F 60 86? , where m is the number minutes it has been cooling. (b) Determine the temperature of the coffee after one da using our calculator. What do ou think this temperature represents about the phsical situation? 8. The graph below shows two eponential functions, with real number constants a, b, c, and d. Given the graphs, onl one pair of the constants shown below could be equal in value. Determine which pair could be equal and eplain our reasoning. b and d a and b a and c a b c d 9. Eplain wh the equation below can have no real solutions. If ou need to, graph both sides of the equation using our calculator to visualize the reason. 5 Unit Lesson 6 Eponential Function Basics Page 91

94 A l g e b r a U n i t - Eponential Functions LESSON 7: FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS In this lesson, ou will learn how to write equations of eponential functions when ou have information about the starting value and base (multiplier or growth constant). Let's review a basic problem. Eercise #1: An eponential function of the form f ab is presented in the table below. Determine the values of a and b and eplain our reasoning. a b 0 1 f Final Equation: Eplanation: Finding an eponential equation becomes more challenging if we do not have output values for inputs that are increasing b 1 unit at a time. Just like with lines, an two points will determine the equation of an eponential function. Steps: 1. Write equations. Divide the equations. Solve for b 4. Solve for a Eercise #: An eponential function of the form (1) B substituting these two points into the general form of the eponential, create a sstem of equations in the constants a and b. ab passes through, 6 and 5,11.5 () Divide these two equations to eliminate the constant a. Recall that when dividing two like bases, ou subtract their eponents. () Solve the resulting equation from () for the base, b. (4) Use our value from () to determine the value of a. State the final equation. Eercise #: For an eponential function of the form f ab, it is known that f Find the values of a and b, and write the equation. f 0 8 and Unit Lesson 7 Finding Equations of Eponential Functions Page 9

95 Steps: 1. Write equations. Divide the equations. Solve for b 4. Solve for a A l g e b r a U n i t - Eponential Functions Eercise #4: An eponential function eists such that f f be the value of its base? Eplain or illustrate our thinking. (1) b 16 () b 6 () b (4) b 4 4 and 6 48, which of the following must Now let's practice this with a decreasing eponential function. Eercise #5: Find the equation of the eponential function shown graphed below. Be careful in terms of our eponent manipulation. State our final answer in the form a b. Eercise #6: A bacterial colon is growing at an eponential rate. It is known that after 4 hours, its population is at 98 bacteria and after 9 hours it is 189 bacteria. Determine an equation in population,, as a function of the number of hours,. (Round to the nearest hundredths.) At what percent rate is the population growing per hour? a b form that models the Unit Lesson 7 Finding Equations of Eponential Functions Page 9

96 A l g e b r a U n i t - Eponential Functions LESSON 7: FINDING EQUATIONS OF EXPONENTIAL FUNCTIONS HOMEWORK 1. For each of the following coordinate pairs, find the equation of the eponential function, in the form a b that passes through the pair. Show the work that ou use to arrive at our answer. (a) 0,10 and, 80 (b) 0,180 and, 80. For each of the following coordinate pairs, find the equation of the eponential function, in the form a b that passes through the pair. Show the work that ou use to arrive at our answer. (a),19 and 5,188 (b) 1,19 and 5, Each of the previous problems had values of a and b that were rational numbers. The do not need not be. Find the equation for an eponential function that passes through the points 7, 05 in,14 and a b form. When ou find the value of b do not round our answer before ou find a. Then, find both to the nearest hundredth and give the final equation. Check to see if the points fall on the curve. Unit Lesson 7 Finding Equations of Eponential Functions Page 94

97 Water Depth (ft) A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) 4. A population of koi goldfish in a pond was measured over time. In the ear 00, the population was recorded as 80 and in 006 it was 517. Given that is the population of fish and is the number of ears since 000, do the following: (a) Represent the information in this problem as two coordinate points. (b) Determine a linear function in the form m b that passes through these two points. Don't round the linear parameters (m and b). (c) Determine an eponential function of the form a b that passes through these two points. Round b to the nearest hundredth and a to the nearest tenth. (d) Which model predicts a larger population of fish in the ear 000? Justif our work. 5. Engineers are draining a water reservoir until its depth is onl 10 feet. The depth decreases eponentiall as shown in the graph below. The engineers measure the depth after 1 hour to be 64 feet and after 4 hours to be 8 feet. Develop an eponential equation in a b to predict the depth as a function of hours draining. Round a to the nearest integer and b to the nearest hundredth. Then, graph the horizontal line 10 and find its intersection to determine the time, to the nearest tenth of an hour, when the reservoir will reach a depth of 10 feet. Time (hrs) Unit Lesson 7 Finding Equations of Eponential Functions Page 95

98 A l g e b r a U n i t - Eponential Functions LESSON 8: THE METHOD OF COMMON BASES In this lesson we will look at solving eponential equations using a method known as The Method of Common Bases. If b = b then Eercise #1: Solve each of the following simple eponential equations b writing each side of the equation using a common base. 1 (a) 16 (b) 7 (c) 5 (d) In each of these cases, even the last, more challenging one, we could manipulate the right-hand side of the equation so that it shared a common base with the left-hand side of the equation. We can eploit this fact b manipulating both sides so that the have a common base. First, though, we need to review an eponent law. Eercise #: Simplif each of the following eponential epressions. (a) (b) 4 1 (c) 7 5 (d) 4 1 Eercise #: Solve each of the following equations b finding a common base for each side. (a) 8 (b) (c) 4 Eercise #4: Which of the following represents the solution set to the equation (1) () 11 0, (4) 5 () 64? Unit Lesson 8 The method of common bases Page 96

99 A l g e b r a U n i t - Eponential Functions This technique can be used in an situation where all bases involved can be written with a common base. In a practical sense, this is rather rare. Yet, these tpes of algebraic manipulations help us see the structure in eponential epressions. Tr to tackle the net, more challenging, problem Eercise #5: Two eponential curves, and are shown below. The intersect at point A. A rectangle has one verte at the origin and the other at A as shown. We want to find its area. 1 (a) Fundamentall, what do we need to know about a rectangle to find its area? (b) How would knowing the coordinates of point A help us find the area? A (c) Find the area of the rectangle algebraicall using the Method of Common Bases. Show our work carefull. Eercise #6: At what coordinate will the graph of the work that leads to our choice. 5 a intersect the graph of ? Show (1) () 5a 1 () a (4) 11 a 1 5 5a Unit Lesson 8 The method of common bases Page 97

100 A l g e b r a U n i t - Eponential Functions LESSON 8: THE METHOD OF COMMON BASES HOMEWORK 1. Solve each of the following eponential equations using the Method of Common Bases. Check our answers. (a) 5 9 (b) 7 16 (c) (d) 1 (e) (f) Algebraicall determine the intersection point of the two eponential functions shown below. 8 and Algebraicall determine the zeroes of the eponential function it is known as a zero is because the output is zero. f. Recall that the reason Unit Lesson 8 The method of common bases Page 98

101 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) 4. One hundred must be raised to what power in order to be equal to a million cubed? Solve this problem using the Method of Common Bases. Show the algebra ou do to find our solution. 5. The eponential function intersection point is a is shown graphed along with the horizontal line 115. Their 5. Use the Method of Common Bases to find the value of a. Show our work.,115 a, The Method of Common Bases works because eponential functions are one-to-one, i.e. if the outputs are the same, then the inputs must also be the same. This is what allows us to sa that if, then must be equal to. But it doesn't alwas work out so easil. If 5, can we sa that must be 5? Could it be anthing else? Wh does this not work out as easil as the eponential case? Unit Lesson 8 The method of common bases Page 99

102 A l g e b r a U n i t - Eponential Functions LESSON 9: EXPONENTIAL MODELING WITH PERCENT RATE GROWTH AND DECAY Eponential functions are ver important in modeling a variet of real world phenomena because certain things either increase or decrease b fied percentages or rates over given units of time. Eercise #1: Suppose that ou deposit mone into a savings account that receives 5% interest per ear on the amount of mone that is in the account for that ear. Assume that ou deposit $400 into the account initiall. (a) How much will the savings account increase b over the course of the ear? (b) How much mone is in the account at the end of the ear? (c) B what single number could ou have multiplied the $400 b in order to calculate our answer in part (b)? (d) Using our answer from part (c), determine the amount of mone in the account after and 10 ears. Round all answers to the nearest cent when needed. (e) Give an equation for the amount in the savings account as a function of the number of ears since the $400 was invested. (f) Using a table on our calculator determine, to the nearest ear, how long it will take for the initial investment of $400 to double. Provide evidence to support our answer. The thinking process from Eercise #1 can be generalized to an situation where a quantit is increased or decreased b a fied percentage over a fied interval of time. This pattern is summarized as follows Eercise #: Which of the following gives the savings S in an account if $50 was invested at an interest rate of % per ear? (1) S 504 t () S () S t (4) S EXPONENTIAL MODELS Some real-life quantities increase or decrease b a fied percent, r, in decimal form. The amount A of such a quantit after t time periods (e.g. ears, minutes, etc.) can be modeled b Eponential Growth Eponential Deca A = a(1 + r) t A = a(1 r) t where a represents the initial amount (amount at t t ) and t represents time. Unit Lesson 9 Eponential Modeling with % Growth and Deca Page 100

103 A l g e b r a U n i t - Eponential Functions Eercise # In 000 the world population was about 6.09 billion. During the net several ears, the world population increased b about 1.18% each ear. a. Write an eponential growth model giving the population A (in billions) t ears after 000. Then, estimate the world population in 005. b. Estimate the ear when the world population was 7 billion. Eercise #4: If the population of a town is decreasing b 4% per ear and started with 1,500 residents, which of the following is its projected population in 10 ears? Show the eponential model ou use to solve this problem. (1) 9,0 () 18,50 () 76 (4) 8,10 Eercise #5: The stock price of WindpowerInc is increasing at a rate of 4% per week. Its initial value was $0 per share. On the other hand, the stock price in GerbilEnerg is crashing (losing value) at a rate of 11% per week. If its price was $10 per share when Windpower was at $0, after how man weeks will the stock prices be the same? Model both stock prices using eponential functions. Then, graphicall find when the stock prices will be equal. Draw a well labeled graph to justif our solution. Eercise #6: State the multiplier (base) ou would need to multipl b in order to decrease a quantit b the given percent listed. (a) 10% (b) % (c) 5% (d) 0.5% Unit Lesson 9 Eponential Modeling with % Growth and Deca Page 101

104 A l g e b r a U n i t - Eponential Functions LESSON 9: EXPONENTIAL MODELING WITH PERCENT GROWTH AND DECAY HOMEWORK 1. If $10 is invested in a savings account that earns 4% interest per ear, which of the following is closest to the amount in the account at the end of 10 ears? (1) $18 () $168 () $19 (4) $4. A population of 50 fruit flies is increasing at a rate of 6% per da. Which of the following is closest to the number of das it will take for the fruit fl population to double? (1) 18 () 1 () 6 (4) 8. If a radioactive substance is quickl decaing at a rate of 1% per hour approimatel how much of a 00 pound sample remains after one da? (1) 7.1 pounds () 5.6 pounds (). pounds (4) 15.6 pounds 4. A population of llamas stranded on a desert island is decreasing due to a food shortage b 6% per ear. If the population of llamas started out at 50, how man are left on the island 10 ears later? (1) 57 () 10 () 58 (4) Which of the following equations would model a population with an initial size of 65 that is growing at an annual rate of 8.5%? (1) t t P () P () P t (4) P 8.5t The acceleration of an object falling through the air will decrease at a rate of 15% per second due to air resistance. If the initial acceleration due to gravit is 9.8 meters per second, which of the following equations best models the acceleration t seconds after the object begins falling? (1) () a a t () a t t (4) a t Unit Lesson 9 Eponential Modeling with % Growth and Deca Page 10

105 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) 7. Red Hook has a population of 6,00 people and is growing at a rate of 8% per ear. Rhinebeck has a population of 8,750 and is growing at a rate of 6% per ear. In how man ears, to the nearest ear, will Red Hook have a greater population than Rhinebeck? Show the equation or inequalit ou are solving and solve it graphicall. 8. A warm glass of water, initiall at 10 degrees Fahrenheit, is placed in a refrigerator at 4 degrees Fahrenheit and its temperature is seen to decrease according to the eponential function h T h (a) Verif that the temperature starts at 10 degrees Fahrenheit b evaluating T 0. (b) Using our calculator, sketch a graph of T below for all values of h on the interval 0 h 4. Be sure to label our -ais and - intercept. (c) After how man hours will the temperature be at 50 degrees Fahrenheit? State our answer to the nearest hundredth of an hour. Illustrate our answer on the graph our drew in (b). 9. Percents combine in strange was that don't seem to make sense at first. It would seem that if a population grows b 5% per ear for 10 ears, then it should grow in total b 50% over a decade. But this isn't true. Start with a population of 100. If it grows at 5% per ear for 10 ears, what is its population after 10 ears? What percent growth does this represent? Unit Lesson 9 Eponential Modeling with % Growth and Deca Page 10

106 A l g e b r a U n i t - Eponential Functions LESSON 10: r-values FOR MULTIPLE OR FRACTIONAL TIME PERIODS t Given the eponential growth formula A a( 1 r) This is an annual growth rate. (Multipl b 100 if ou want the annual percent growth rate.), when t = 1 (e.g. 1 ear), solve for r. Finding the annual rate of growth (find r when t=1) Eercise #1: A population of wombats is growing at a constant percent rate. If the population on Januar 1 st is 107 and a ear later is 1079, what is its annual percent growth rate to the nearest tenth of a percent? Finding the percent growth over multiple ears (when t>1) Eercise #: Now let's determine the percent growth in wombat population over a decade. Assume the rounded annual percent increase found in Eercise #1 continues for the net decade. (a) After 10 ears, what will the original population be multiplied b, rounded to the nearest hundredth? Show the calculation. (b) Using our answer from (a), what is the decade percent growth rate? Finding the growth rate for a fraction of the ear (when 0<t<1, in eercises and 4) Eercise #: Use the wombats from Eercise #1. Assuming their annual growth rate is constant, what is the monthl growth rate to the nearest tenth of a percent? Assume a constant sized month. Eercise #4: If a population grows at a constant rate of % ever 5 ears, what is its percent growth rate over a ear time span? Round to the nearest tenth of a percent. (a) First, give an epression that will calculate the earl percent growth rate based on the fact that the population grew % in 5 ears. (b) Now use this epression to calculate the percent growth over ears. Unit Lesson 10 r-values for multiple or fractional time periods Page 104

107 A l g e b r a U n i t - Eponential Functions Eercise #5: World oil reserves (the amount of oil unused in the ground) are depleting at a constant % per ear. Determine the percent decline over the net 0 ears based on this % earl decline. (a) Write and evaluate an epression for what we would multipl the initial amount of oil b after 0 ears. (b) Use our answer to (a) to determine the percent decline, r, after 0 ears. Be careful! Round to the nearest percent. Eercise #6: The population of squirrels in Ulster Count is growing at an annual rate of 1.8%. Find the percent rate of growth (a) ever ears (b) ever 5 ears (c) ever Decade (d) Monthl Show the calculations that lead to each answer. Round each to the nearest tenth of a percent. Eercise #7: A radioactive substance s half-life is the amount of time needed for half (or 50%) of the substance to deca. A certain radioactive substance has a half-life of 15 ears. (a) First, give an epression that will calculate the earl percent deca rate based on the fact that the substance decaed 50% in 15 ears. (b) What percent of the substance would be radioactive after 60 ears? (c) What percent of the substance would be radioactive after 5 ears? Round to the nearest tenth of a percent. Eercise #8: Rewrite the following in the form A = a(1 ± r) t. State the growth or deca rate r. a) A = a() t/ b) A = a(4) t/6 c) A = a(.5) t/1 d) A = a(.5) t/9 Unit Lesson 10 r-values for multiple or fractional time periods Page 105

108 A l g e b r a U n i t - Eponential Functions LESSON 10: r-values FOR MULTIPLE OR FRACTIONAL TIME PERIODS HOMEWORK 1. A quantit is growing at a constant % earl rate. Which of the following would be its percent growth after 15 ears? (1) 45% () 56% () 5% (4) 6%. If a credit card compan charges 1.5% earl interest, which of the following calculations would be used in the process of calculating the monthl interest rate? (1) () () (4) The count debt is growing at an annual rate of.5%. What percent rate is it growing ever ears? Ever 5 ears? Ever decade? Show the calculations that lead to each answer. Round each to the nearest tenth of a percent. 4. A population of llamas is growing at a constant earl rate of 6%. At what rate is the llama population growing per month? Assume all months are equall sized and there are 1 of these per ear. Round to the nearest tenth of a percent. Unit Lesson 10 r-values for multiple or fractional time periods Page 106

109 A l g e b r a U n i t - Eponential Functions HOMEWORK (cont.) 5. Shana is tring to increase the number of calories she burns b 5% per da. B what percent is she tring to increase per week? Round to the nearest tenth of a percent. 6. If a bank account doubles in size ever 5 ears, then b what percent does it grow after onl ears? Round to the nearest tenth of a percent. Hint: First write an epression that would calculate its growth rate after a single ear. 7. An object s speed decreases b 5% for each minute that it is slowing down. Which of the following is closest to the percent that its speed will decrease over half-an hour? (1) 1% () 48% () 79% (4) 150% 8. Over the last 10 ears, the price of corn has decreased b 5% per bushel. (a) Assuming a stead percent decrease, b what percent does it decrease each ear? Round to the nearest tenth of a percent. (b) Assuming this percent continues, b what percent will the price of corn decrease b after 50 ears? Show the calculation that leads to our answer. Round to the nearest percent. Unit Lesson 10 r-values for multiple or fractional time periods Page 107

110 A l g e b r a U n i t 4 - Logarithmic Functions U n i t 4 - Logarithmic Functions LESSON 1 - INTRODUCTION TO LOGARITHMS Eponential functions are of such importance that their inverses, functions that reverse their action, are important themselves. These functions, known as logarithms, will be introduced in this lesson. Eercise #1: The function f is shown graphed on the aes below along with its table of values f (a) Is this function one-to-one? Eplain our answer. (b) Based on our answer from part (a), what must be true about the inverse of this function? (c) Create a table of values below for the inverse of f and plot this graph on the aes given. Notice that, as alwas, the graphs of and are smmetric across f 1 (d) What would be the first step to find an equation for this inverse algebraicall? Write this step down and then stop. Defining Logarithmic Functions The function log b is the name we give the inverse of b. For eample, is the inverse of. Based on Eercise #1(d), we can write an equivalent eponential equation for each logarithm as follows: log is the same as b b Based on this, we see that a logarithm gives as its output (-value) the eponent we must raise b to in order to produce its input (-value). Unit 4 Lesson 1 Introduction to Logarithms Page 108

111 A l g e b r a U n i t 4 - Logarithmic Functions It is criticall important to understand that logarithms give eponents as their outputs. Eercise #: Evaluate the following logarithms. If needed, write an equivalent eponential equation. Do as man as possible without the use of our calculator. (a) log 8 (b) log 416 (c) log5 65 (d) log10100,000 (e) log6 1 6 (f) log (g) log5 5 (h) log 9 Eercise #: If the function would represent its -intercept? (1) 1 () 8 () 1 (4) 9 log 8 9 was graphed in the coordinate plane, which of the following Eercise #4: Between which two consecutive integers must log 40 lie? (1) 1 and () and 4 () and (4) 4 and 5 Calculator Use and Logarithms Most calculators onl have two logarithms that the can evaluate directl. One of them, log10, is so common that it is actuall called the common log and tpicall is written without the base 10. log log 10 (The Common Log) Eercise #5: Evaluate each of the following using our calculator. (a) log100 (b) log (c) log 10 Switching between eponential form and log form. Eponential Form Log Form Eponential Form Log Form ³ = 8 log 8 log = log log h p r s t m log log log b 9 log Unit 4 Lesson 1 Introduction to Logarithms Page 109

112 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 1 - INTRODUCTION TO LOGARITHMS HOMEWORK 1. Which of the following is equivalent to log7? (1) 7 () 7 () 7 (4) 1 7. If the graph of 6 is reflected across the line then the resulting curve has an equation of (1) 6 () log6 () log6 (4) 6. The value of log5 167 is closest to which of the following? Hint guess and check the answers. (1).67 () 4.58 () 1.98 (4) Which of the following represents the -intercept of the function (1) 8 () () 5 (4) 5 log ? 5. Determine the value for each of the following logarithms. (Eas) (a) log (b) log7 49 (c) log 6561 (d) log Determine the value for each of the following logarithms. (Medium) (a) log 1 64 (b) log 1 (c) log5 1 5 (d) log7 1 4 Unit 4 Lesson 1 Introduction to Logarithms Page 110

113 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 7. Determine the value for each of the following logarithms. Each of these will have non-integer, fractional answers. (Difficult) 5 (a) log 4 (b) log 48 (c) log 5 (d) log Between what two consecutive integers must the value of log 4 74 lie? Justif our answer. 9. Between what two consecutive integers must the value of log lie? Justif our answer. 10. In chemistr, the ph of a solution is defined b the equation ph log H where H represents the concentration of hdrogen ions in the solution. An solution with a ph less than 7 is considered acidic and an solution with a ph greater than 7 is considered basic. Fill in the table below. Round our ph s to the nearest tenth of a unit. Substance Concentration of Hdrogen 7 Milk Coffee Bleach.5 10 Lemon Juice Rain ph Basic or Acidic? 11. Can the value of log 4 tell ou about the domain of log b? be found? What about the value of log 0? Wh or wh not? What does this Unit 4 Lesson 1 Introduction to Logarithms Page 111

114 A l g e b r a U n i t 4 - Logarithmic Functions LESSON - GRAPHS OF LOGARITHMS Most logarithms have bases greater than one; the ph scale that we saw on the last homework assignment is a good eample. In this lesson, we will further eplore graphs of these logarithms, including their construction, transformations, and domains and ranges. Eercise #1: Consider the logarithmic function log and its inverse. (a) Construct a table of values for and then use this to construct a table of values for the function log log (b) Graph and log on the grid given. Label with equations. Label the asmptote and its equation. (c) State the domain and range of and log. Write the equation of the asmptote for each. Domain: Range: Domain: Range: log Asmptote: Asmptote: Eercise #: Using our calculator, sketch the graph of log10 on the aes below. Label the -intercept. State the domain and range of log10. Domain: Range: Equation of asmptote: 10 Unit 4 Lesson Graphs of Logarithms Page 11

115 A l g e b r a U n i t 4 - Logarithmic Functions Eercise #: Which of the following equations describes the graph shown below? Show or eplain how ou made our choice. (1) log 1 () log 1 () log 1 (4) log 1 The fact that finding the logarithm of a non-positive number (negative or zero) is not possible in the real number sstem allows us to find the domains of a variet of logarithmic functions. Eercise #4: Determine the domain of the following functions. State our answer in set-builder notation. (a) log 4 (b) = log ( + 4) (c) = log 5 (7 ) All logarithms with bases larger than 1 are alwas increasing. This increasing nature can be seen b calculating their average rate of change. Eercise #5: Consider the common log, or log base 10, f log. (a) Set up and evaluate an epression for the average rate of change of f over the interval 1 10 (b) Set up and evaluate an epression for the average rate of change of f over the interval Unit 4 Lesson Graphs of Logarithms Page 11

116 1. The domain of A l g e b r a U n i t 4 - Logarithmic Functions LESSON - GRAPHS OF LOGARITHMS HOMEWORK log 5 in the real numbers is (1) 0 () 5 () 5 (4) 4. Which of the following equations describes the graph shown below? (1) log5 () log () log (4) log 4. Which of the following represents the -intercept of the function (1) 8 () 1 () 4 (4) 4 log 1? 4. Which of the following values of is not in the domain of f log 10 (1) () 5 () 0 (4) 4 5. Which of the following is true about the function (1) It has an -intercept of 4 and a -intercept of 1. () It has -intercept of 1 and a -intercept of 1. () It has an -intercept of 16 and a -intercept of 1. (4) It has an -intercept of 16 and a -intercept of 1. log 16 1? 4 5? 6. The graph of the function = log 5 appears in which quadrants? (a) I and II (b) I and IV (c) II and III (d) III and IV Unit 4 Lesson Graphs of Logarithms Page 114

117 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 7. Determine the domains of each of the following logarithmic functions. State our answers using an accepted notation. Be sure to show the inequalit that ou are solving to find the domain and the work ou use to solve the inequalit. (a) log 1 (b) log Graph the logarithmic function log 4 on the graph paper given. For a method, see Eercise #1. 9. Logarithmic functions whose bases are larger than 1 tend to increase ver slowl as increases. Let's investigate this for f log. (a) Find the value of f 1, f, f 4, and f 8 without our calculator. (b) For what value of will? For what value of will log 10 log 0? 10. If the graph of 6 is reflected across the line then the resulting curve has an equation of (1) 6 () log6 () log6 (4) 6 Unit 4 Lesson Graphs of Logarithms Page 115

118 A l g e b r a U n i t 4 - Logarithmic Functions LESSON - SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS Earlier in this unit, we used the Method of Common Bases to solve eponential equations. This technique is quite limited, however, because it requires the two sides of the equation to be epressed using the same base. A more general method utilizes our calculators. Eercise #1: Solve: 4 8 using (a) common bases and (b) the logarithm law shown above. (a) Method of Common Bases (b) Logarithm Approach The beaut of using logarithms is that it removes the variable from the eponent. We can solve almost an eponential equation using a TI-84 calculator as follows: alpha window #5. Eercise #: Solve each of the following equations for the value of. Round our answers to the nearest hundredth. **You must isolate the base before switching to log form!** (a) 5 18 (b) 4 +1 = 11 (c) These equations can become more complicated, but each and ever time we will use the logarithm law to transform an eponential equation into one that is more familiar (linear onl for now) Eercise #: Solve each of the following equations for. Round our answers to the nearest hundredth. (a) 6 50 (b) Unit 4 Lesson Solving Eponential Equations using Logs Page 116

119 A l g e b r a U n i t 4 - Logarithmic Functions Now that we are familiar with this method, we can revisit some of our eponential models from earlier in the unit. Recall that for an eponential function that is growing: If quantit A is known to increase b a fied percentage r, in decimal form, then A can be modeled b where P represents the amount of A present at A(t) = P(1 + r) t and t represents time. Eercise #4: A biologist is modeling the population of bats on a tropical island. When he first starts observing them, there are 104 bats. The biologist believes that the bat population is growing at a rate of % per ear. (a) Write an equation for the number of bats,, as a function of the number of ears, t, since the biologist started observing them. (b) Using our equation from part (a), algebraicall determine the number of ears it will take for the bat population to reach 00. Round our answer to the nearest ear. Eercise #5: A stock has been declining in price at a stead pace of 5% per week. If the stock started at a price of $.50 per share, determine algebraicall the number of weeks it will take for the price to reach $ Round our answer to the nearest week. Eercise #6: Solve each of the following eponential equations to the nearest hundredth. (a) 4 17 (b) Unit 4 Lesson Solving Eponential Equations using Logs Page 117

120 A l g e b r a U n i t 4 - Logarithmic Functions LESSON - SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMS HOMEWORK 1. Solve for. When necessar, round our answer to the nearest ten thousandth. (4 decimal places) (a) 7 8 (b) log 4 (c) 4 1 (d) log Solve each of the following eponential equations. Round each of our answers to the nearest hundredth. (a) 9 50 (b) (c) Solve each of the following eponential equations. Be careful with our use of parentheses. Epress each answer to the nearest hundredth. (a) (b) (c) (d) 5(10) 6 = 100 (e) 1() 4 =117 Unit 4 Lesson Solving Eponential Equations using Logs Page 118

121 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 4. The population of Red Hook is growing at a rate of.5% per ear. If its current population is 1,500, in how man ears will the population eceed 0,000? Round our answer to the nearest ear. Onl an algebraic solution is acceptable. 5. A radioactive substance is decaing such that % of its mass is lost ever ear. Originall there were 50 kilograms of the substance present. (a) Write an equation for the amount, A, of the substance left after t-ears. (b) Find the amount of time that it takes for onl half of the initial amount to remain. Round our answer to the nearest tenth of a ear. 6. If a population doubles ever 5 ears, how man ears will it take for the population to increase b 10 times its original amount? First: If the population gets multiplied b ever 5 ears, what does it get multiplied b each ear? Use this to help ou answer the question. 7. Find the solution to the general eponential equation a b c d, in terms of the constants a, c, d and the logarithm of base b. Think about reversing the order of operations in order to solve for. Unit 4 Lesson Solving Eponential Equations using Logs Page 119

122 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 4 - THE NUMBER e AND THE NATURAL LOGARITHM There are man numbers that are more important than others because the find so man uses in either math or science. Good eamples of important numbers are 0, 1, i, and. In this lesson ou will be introduced to an important number given the letter e for its inventor Leonhard Euler ( ). This number plas a crucial role in Calculus and more generall in modeling eponential phenomena. THE NUMBER e 1. Like, e is irrational.. e. Used in Eponential Modeling Eercise #1: Which of the graphs below shows e? Eplain our choice. Check on our calculator. (1) () () (4) Eplanation: Ver often e is involved in eponential modeling of both increasing and decreasing quantities. Eercise #: A population of llamas on a tropical island can be modeled b the equation t represents the number of ears since the llamas were first introduced to the island. (a) How man llamas were initiall introduced at t 0? Show the calculation that leads to our answer. P 500e 0.05t, where (b) Algebraicall determine the number of ears for the population to reach 600. Round our answer to the nearest tenth of a ear. Unit 4 Lesson 4 The number e and the Natural Logarithm Page 10

123 A l g e b r a U n i t 4 - Logarithmic Functions Because of the importance of e, its inverse, known as the natural logarithm, is also important. The natural logarithm, like all logarithms, gives an eponent as its output. In fact, it gives the power that we must raise e to in order to get the input. Eercise #: Without the use of our calculator, determine the values of each of the following. (a) lne (b) THE NATURAL LOGARITHM The inverse of : 5 ln 1 (c) ln e (d) ln e Solve for to the nearest hundredth: (e) ln() =.5787 (f) e 5 The natural logarithm follows the basic logarithm laws that all logarithms follow. The following problems give additional practice with these laws. Eercise #4: A hot liquid is cooling in a room whose temperature is constant. Its temperature can be modeled using the eponential function shown below. The temperature, T, is in degrees Fahrenheit and is a function of the number of minutes, m, it has been cooling. 0.0 e T m m (a) What was the initial temperature of the water at m 0. Do without using our calculator. (b) How do ou interpret the statement that T ? (c) Using the natural logarithm, determine algebraicall when the temperature of the liquid will reach 100 F. Show the steps in our solution. Round to the nearest tenth of a minute. (d) On average, how man degrees are lost per minute over the interval 10 m 0? Round to the nearest tenth of a degree. Unit 4 Lesson 4 The number e and the Natural Logarithm Page 11

124 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 4 - THE NUMBER e AND THE NATURAL LOGARITHM HOMEWORK 1 10e? 1. Which of the following is closest to the -intercept of the function whose equation is (1) 10 () 7 () 18 (4) 5. On the grid below, the solid curve represents e. Which of the following eponential functions could describe the dashed curve? Eplain our choice. (1) 1 () () e (4) 4. Which of the following values of t solves the equation (1) ln15 10 () 1 ln 5 () ln (4) ln t 5 15 e? 4. At which of the following values of does (1) 5 ln () ln8 f e have a zero? () ln 4 (4) ln 5 ct 5. For the equation ae d, solve for the variable t in terms of a, c, and d. Epress our answer in terms of the natural logarithm. Unit 4 Lesson 4 The number e and the Natural Logarithm Page 1

125 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 6. Flu is spreading eponentiall at a school. The number of new flu patients can be modeled using the 0.1d equation F 10e, where d represents the number of das since 10 students had the flu. (a) How man das will it take for the number of new flu patients to equal 50? Determine our answer algebraicall using the natural logarithm. Round our answer to the nearest da. (b) Find the average rate of change of F over the first three weeks, i.e. 0 d 1. Show the calculation that leads to our answer. Give proper units and round our answer to the nearest tenth. What is the phsical interpretation of our answer?.045t 7. The savings in a bank account can be modeled using S 150e, where t is the number of ears the mone has been in the account. Determine, to the nearest tenth of a ear, how long it will take for the amount of savings to double from the initial amount deposited of $ Solve for to the nearest thousandth: (a) 80 (b) ln () =.55 Unit 4 Lesson 4 The number e and the Natural Logarithm Page 1

126 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 5 - COMPOUND INTEREST Compound interest is interest paid on an initial investment, called the principal, and on previousl earned interest. Interest earned is often epressed as an annual percent, but the interest is usuall compounded t more than once per ear. So, the eponential growth model A a( 1 r) must be modified for compound interest problems. Compound Interest Given P = amount initiall invested (Principal) r = annual interest rate epressed as a decimal n = number of compounds per ear t = time, or number of ears The amount A in the account after t ears is given b the formula A P 1 r n nt Eercise #1: A person invests $500 in an account that earns a nominal earl interest rate of 4%. Calculate the amount of mone in the account if the interest was compounded as follows: (a) Quarterl (b) Monthl (c) Dail Eercise #: How much would $1000 invested at a nominal % earl rate, compounded monthl, be worth in 0 ears? Show the calculations that lead to our answer. (1) $ () $10.87 () $1491. (4) $1045. Unit 4 Lesson 5 Compound Interest Page 14

127 A l g e b r a U n i t 4 - Logarithmic Functions Eercise #: If $1500 is invested at.5% interest compounded weekl, (a) how much will be in the account after 10 ears to the nearest dollar, (b) how man ears will it take for the account to reach $500? Round to nearest tenth of a ear. Eercise #4: If $100 is invested at 8% interest compounded monthl, after how man ears will the amount in the account double? Round to the nearest tenth of a ear. The rate in Eercise #1 was referred to as nominal (in name onl). It's known as this, because ou effectivel earn more than this rate if the compounding period is more than once per ear. Because of this, bankers refer to the effective rate, or the rate ou would receive if compounded just once per ear. Let's investigate this. Eercise #5: An investment with a nominal rate of 5% is compounded at different frequencies. Give the effective earl rate, accurate to two decimal places, for each of the following compounding frequencies. Show our calculation. (a) Quarterl (b) Monthl (c) Dail Unit 4 Lesson 5 Compound Interest Page 15

128 A l g e b r a U n i t 4 - Logarithmic Functions We could compound at smaller and smaller frequenc intervals, eventuall compounding all moments of time. This gives rise to continuous compounding and the use of the natural base e in the continuous compound interest formula. CONTINUOUS COMPOUND INTEREST For an initial principal, P, compounded continuousl at a nominal earl rate of r, the investment would be worth an amount A given b: A = Pe rt Eercise #6: Tom invests $50 in a bank account that pas % annual interest compounded continuousl. (a) Write an equation for the amount this investment would be worth after t-ears. (b) How much would the investment be worth after 0 ears? (c) Algebraicall determine the time it will take for the investment to reach $400. Round to the nearest tenth of a ear. (d) What is the effective annual rate for this investment? Round to the nearest hundredth of a percent. Eercise #7: You invest $,500 in an account to save for college. Account 1 pas 6% annual interest compounded quarterl. Account pas 4% annual interest compounded continuousl. Which account should ou choose to obtain the greater amount in 10 ears? Justif our answer. The above formula can be applied to an situation that grows at a continuous rate. Eercise #8: A population of 14 seals on a tropical island is growing continuousl at a rate of 1.5% per ear. (a) Write a function to model the number of seals on the island after t-ears. (b) Algebraicall determine the number of ears for the population to reach 455. Round our answer to the nearest tenth of a ear. Unit 4 Lesson 5 Compound Interest Page 16

129 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 5 - COMPOUND INTEREST HOMEWORK 1. The value of an initial investment of $400 at % nominal interest compounded quarterl can be modeled using which of the following equations, where t is the number of ears since the investment was made? (1) A t () A t () A t (4) A t. Which of the following represents the value of an investment with a principal of $1500 with a nominal interest rate of.5% compounded monthl after 5 ears? (1) $1, () $4,178. () $1, (4) $5, Franco invests $4,500 in an account that earns a.8% nominal interest rate compounded continuousl. If he withdraws the profit from the investment after 5 ears, how much has he earned on his investment? (1) $858.9 () $9.50 () $91.59 (4) $ An investment that returns a nominal 4.% earl rate, but is compounded quarterl, has an effective earl rate closest to (1) 4.1% () 4.7% () 4.4% (4) 4.% 5. If an investment's value can be modeled with the investment?.07 A t then which of the following describes (1) The investment has a nominal rate of 7% compounded ever 1 ears. () The investment has a nominal rate of.7% compounded ever 1 ears. () The investment has a nominal rate of 7% compounded 1 times per ear. (4) The investment has a nominal rate of.7% compounded 1 times per ear. Unit 4 Lesson 5 Compound Interest Page 17

130 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 6. An investment of $500 is made at.8% nominal interest compounded quarterl. (a) Write an equation that models the amount A the investment is worth t-ears after the principal has been invested. (b) How much is the investment worth after 10 ears? (c) Algebraicall determine the number of ears it will take for the investment to reach a worth of $800. Round to the nearest hundredth. (d) Wh does it make more sense to round our answer in (c) to the nearest quarter? State the final answer rounded to the nearest quarter. 7. An investment of $00 is made at.6% nominal interest compounded continuousl. (a) Write an equation that models the amount A the investment is worth t-ears after the principal has been invested. (b) How much is the investment worth after 10 ears? (c) Algebraicall determine the number of ears it will take for the investment to be reach a worth of $800. Round to the nearest hundredth. rt 8. The formula A Pe calculates the amount an investment earning a nominal rate of r compounded continuousl is worth. Show that the amount of time it takes for the investment to double in value is given b the epression ln. r Unit 4 Lesson 5 Compound Interest Page 18

131 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 6 - MORE EXPONENTIAL AND LOGARITHMIC MODELING The Half-life, h, is the amount of time required for the amount of something to decrease to half its initial value. An eponential deca function can be rewritten as a half-life function. We will begin with an eample from the first lesson of this unit: Eercise #1: The population of a town is decreasing b 4% per ear and started with 1,500 residents. (a) Write a function to model this situation. (b) Algebraicall determine when the population of the town will be half of the initial population. Round to the nearest tenth of a ear. The time ou calculate is called the half-life, or h. To re-write (a) as a half-life formula, start with the eponential deca formula A = a(1 r) t. Since the amount decreases to half its amount, the r value is 50%, i.e., r = 1 over the timespan of the half-life h. 1 1 Thus the annual deca rate is (1 1 ) h, or ( 1 ) h. To calculate the amount A remaining after t ears, raise 1 t this annual deca rate to the t power: (( 1 ) h ). Use the power rule of eponents to obtain the following: HALF LIFE FORMULA For an initial quantit, a, that is decreasing at an eponential rate with a half life, h, the amount, A, left after t time units is given b the formula A = a( 1 t ) h (c) Write the half-life formula for the population of the town for Eercise #1. Eercise #: The deca of a sample of 5000 grams of carbon can be modeled b the equation, t Ct ( ) 5000, where t is measured in ears. (a) What is the half-life of carbon? (b) How can ou tell this is a half-life equation? Unit 4 Lesson 6 - More Eponential and Log Modeling Page 19

132 A l g e b r a U n i t 4 - Logarithmic Functions Eercise #: One of the medical uses of I-11, a radioactive isotope of Iodine, is to enhance -ra images. The half-life of I-1 is 8.0 das. A patient is injected with 0 milligrams of I-11. Determine, to the nearest da, the amount of time needed before the amount of I-11 in the patient s bod is 7 milligrams. Log functions can be used to model real world phenomena. Eample #4: The slope, s, of a beach is related to the average diameter d (in millimeters) of the sand particles on the beach b this equation: s log d. s (a) Sand particles tpicall have a maimum diameter of 1mm. Using this information, sketch a graph of the function. (b) If the average diameter of the sand particles is 0.5mm, find the slope of the beach (to the nearest hundredth). (c) Given a slope of 0.14, find the average diameter (to the nearest hundredth) of the sand particles on the beach. d Eample #5: Two methods were used to teach athletes how to shoot a basketball. The methods were assessed b assigning students into groups, one group taught with method A, and one group taught with method B. The students in each group took 0 foul shots after ten sessions. The average number of shots made in each of the sessions using method A can be modeled b A( ) ln. The average number of shots made in each of the sessions using method B can be modeled b B ( ) 9.17(1.109). (a) In which of the 10 sessions, to the nearest whole number, will the two methods produce the same number of made baskets? Eplain how ou found our answer. (b) Find the average range of change for each method between sessions and 8 for each of the methods. Give proper units and round our answers to the nearest tenth. (c) Eplain wh B() would not be an appropriate model for this situation if there were 15 sessions. Unit 4 Lesson 6 - More Eponential and Log Modeling Page 10

133 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 6 - MORE EXPONENTIAL AND LOGARITHMIC MODELING HOMEWORK 1. The deca of a sample of 800 grams of hdrogen can be modeled b the equation, t ears. (a) What is the half-life of hdrogen? t 1 Ht ( ) (b) How can ou tell this is a half-life equation? after. The $,500 in our bank account is decreasing continuousl at a rate of 5% per ear. (a) Use the Continuous Compound Interest Formula to write a function that models the amount of mone in our bank account after t ears. (Don t forget the r value should be negative because it s decreasing). (b) When will onl half of our initial deposit be left in our bank account (to the nearest tenth of a ear)? (c) Write the half-life formula for our bank account.. Flu is spreading eponentiall at a school. The number of new flu patients can be modeled using 0.1d the equation F 10e, where d represents the number of das since 10 students had the flu. (a) How man das will it take for the number of new flu patients to equal 50? Round our answer to the nearest da. (b) Find the average rate of change of F over the first three weeks, i.e. 0 d 1. Show the calculation that leads to our answer. Give proper units and round our answer to the nearest tenth. What is the phsical interpretation of our answer? Unit 4 Lesson 6 - More Eponential and Log Modeling Page 11

134 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 4. Jessica keeps track of the height of a tree she planted over the first ten ears. It can be modeled b the equation ln( 1) where is the number of ears since she planted the tree. (a) On average, how man feet did the tree grow each ear over the time interval 0 t 10, to the nearest hundredth. (b) How tall was the tree when she planted it? 5. Most tornadoes last less than an hour and travel less than 0 miles. The wind speed s (in miles per hour) near the center of a tornado is related to the distance d (in miles) the tornado travels b this model: s = 9logd +65. (a) Sketch a graph of this function. (b) On March 18, 195, a tornado whose wind speed was about 180 miles per hour struck the Midwest. Use our graph to determine how far the tornado traveled to the nearest mile. Unit 4 Lesson 6 - More Eponential and Log Modeling Page 1

135 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 7: NEWTON'S LAW OF COOLING AND EXPONENTIAL FORMULA REVIEW NEWTON S LAW OF COOLING where: T(t) is the temperature of the object t time units has elapsed T a is the ambient temperature (the temperature of the surroundings), assumed to be constant T 0 is the initial temperature of the object k is the deca constant per time unit (the r value where r is negative). Eercise #1: A detective is called to the scene of a crime where a dead bod has just been found. He arrives at the scene and measures the temperature of the dead bod at 9:0 p.m to be 7 F. After investigating the scene, he declares that the person died 10 hours prior, at approimatel 11:0 a.m. A crime scene investigator arrives a little later and declares that the detective is wrong. She sas that the person died at approimatel 6:00 a.m., hours prior to the measurement of the bod temperature. She claims she can prove it b using Newton s law of cooling. Using the data collected at the scene, decide who is correct, the detective or the crime scene investigator. T a = 68 F (the temperature of the room) T 0 = 98.6 F (the initial temperature of the bod) k = 0.15 (1.5 % per hour calculated b the investigator from the data collected) Recall, the temperature of the bod at 9:0 p.m. is 7 F. Eercise : A detective is called to the scene of a crime where a dead bod has just been found. She arrives on the scene at 10: pm and begins her investigation. Immediatel, the temperature of the bod is taken and is found to be 80 o F. The detective checks the programmable thermostat and finds that the room has been kept at a constant 68 o F. Assuming that the victim s bod temperature was normal (98.6 o F) prior to death and that the temperature of the victim s bod decreases continuousl at a rate of 1.5% per hour, use Newton s Law of Cooling to determine the time when the victim died. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 1

136 A l g e b r a U n i t 4 - Logarithmic Functions Eercise #: Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 180 F and k is 0.7 (for time in minutes). Suppose both cups are poured at the same time. Cup 1 is left sitting in the room that is 75 F, and cup is taken outside where it is 4 F. i. Use Newton s law of cooling to write equations for the temperature of each cup of coffee after t minutes have elapsed. ii. Graph and label both on the same coordinate plane and compare and contrast the end behavior of the two graphs. iii. Coffee is safe to drink when its temperature is below 140 F. How much time elapses before each cup is safe to drink, to the nearest tenth of a minute. Use a graph to answer the question. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 14

137 A l g e b r a U n i t 4 - Logarithmic Functions Throughout the unit, ou have learned man different eponential formulas. We will now practice writing a few of them and converting between the different forms. Eercise #4: Tritium has a half-life of 1. ears. a. Write a half-life formula, A(t), for the amount of tritium left in a 500 milligram sample after t ears. b. Write an equivalent function, B(t), it terms of the earl rate of deca of tritium. Round all values to four decimal places. c. Wrtie an equivalent function, C(t), it terms of the monthl rate of deca of tritium. Round all values to four decimal places. Eercise #5: A deposit of $00 is made into a bank account that gets 4.% interest compounded continuousl. a. Write a function, A(t), to model the amount of mone in the account after t ears. b. Write an equivalent function, B(t), it terms of the earl rate of interest for the account. Round all values to four decimal places. c. Write an equivalent function, C(t), it terms of the quarterl rate of interest for the account. Round all values to four decimal places. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 15

138 A l g e b r a U n i t 4 - Logarithmic Functions LESSON 7: NEWTON'S LAW OF COOLING AND EXPONENTIAL FORMULA REVIEW HOMEWORK 1. Hot soup is poured from a pot and allowed to cool in a room. The temperature in degrees Fahrenheit of the soup after t minutes, can be modeled b the function, T(t)= 65+(1-65)e -.054t. What was the initial temperature of the soup? What is the temperature of the room? At what rate is the temperature of the soup decreasing?. Two cups of coffee are poured from the same pot. The initial temperature of the coffee is 190 F and k is (for time in minutes). Both are left sitting in the room that is 75 F, but milk is immediatel poured into cup cooling it to an initial temperature of 16 F. a. Use Newton s law of cooling to write equations for the temperature of each cup of coffee after t minutes have elapsed. b. Graph and label both functions on the coordinate plane and compare and contrast the end behavior of the two graphs. c. Coffee is safe to drink when its temperature is below 140 F. Based on our graph, how much time elapses before each cup is safe to drink to the nearest tenth of a minute? Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 16

139 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.). A cooling liquid starts at a temperature of 00 F and cools down in a room that is held at a constant temperature of 70 F. (Note time is measured in minutes on this problem). (a) Use Newton s Law of Cooling to determine the value of k if the temperature after 5 minutes is. Round to four decimal places. (Hint: Write out the equation, plug in (5,15), and solve for k.) (b) Using the value of k ou found in part (a), algebraicall determine, to the nearest tenth of a minute, when the temperature reaches 100 F. 4. A deposit of $100 is made into a bank account that gets 4.% interest compounded weekl. a. Write a function, A(t), to model the amount of mone in the account after t ears. b. Write an equivalent function, B(t), it terms of the earl rate of interest for the account. Round all values to four decimal places. c. Write an equivalent function, C(t), it terms of the monthl rate of interest for the account. Round all values to four decimal places. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 17

140 A l g e b r a U n i t 4 - Logarithmic Functions HOMEWORK (cont.) 5. A small town has a population of 1,600. The population is decreasing continuousl at a rate of % per ear. a. Write a function, A(t), to model the population of the town after t ears. b. Write an equivalent function, B(t), it terms of the earl rate of decrease for the town. Round all values to four decimal places. c. Write an equivalent function, C(t), it terms of the dail rate of decrease for the town. Round all values to four decimal places. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 18

141 A l g e b r a U n i t 4 - Logarithmic Functions Do now: 1. Which of the following values, to the nearest hundredth, solves: 7 500? (1).19 ().74 ().8 (4).17. The solution to 5, to the nearest tenth, is which of the following? (1) 7. () 11.4 () 9.1 (4) To the nearest hundredth, the value of that solves is (1) 6.7 () 8.17 () 5.74 (4) Growth of a certain strain of bacteria is modeled b the equation G = A(.7) 0.584t, where G = final number of bacteria, A=initial number of bacteria, and t = time (in hours). In approimatel how man hours will 4 bacteria first increase to,500 bacteria? Round our answer to the nearest hour. 5. The rate of bacteria growth on a piece of mold bread is represented b the equation r = 4(.5) t, t representing time in minutes and R representing the amount of bacteria in millions. If there was originall 4 million specimens of bacteria, how man minutes will it take for there to be triple that amount? Round our answer to the nearest tenth of a minute. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 19

142 A l g e b r a U n i t 4 - Logarithmic Functions 6. Hannah invests $,850 at an annual rate of 6% compounded compounded continuousl. (a) Determine, to the nearest dollar, the amount of mone she will have after 5 ears. (b) Determine how man ears, to the nearest ear, it will take for her investment to have a value of $10, The deca of a sample of radioactive iodine can be modeled b the function f(t) = 80(. 5) t 60, where f grams of the radioactive element remain after t das. In approimatel how man das will 15% of the original mass be present? 8. The Franklins inherited $,500, which the want to invest for their child s future college epenses. If the invest it at 8.5% with interest compounded monthl, determine the value of the account, in dollars, after 5 ears 9. The Matthews famil would like to invest $,000 for their child s future college epenses. If the invest it at 6.75% with interest compounded monthl, determine the amount of time, to the nearest ear, for the investment to double in value. Unit 4 Lesson 7 - Newton s Law of Cooling and Ep. Formula Review Page 140

143 A l g e b r a U n i t 5 - Sequences and Series U n i t 5 - Sequences and Series LESSON 1 - SEQUENCES Sequences are ordered lists of numbers and are etremel important in math. A sequence is a function whose domain the set of positive integers, i.e. 1,,,..., n. Eercise #1: Given the following eplicit sequence definition: an = an n 1 (a) Find the first three terms of this sequence, written a, a, and a. 1. (b) Find the value of the 40 th term. (c) Which term has a value of 5? (d) Eplain wh there will not be a term that has a value of 70. (e) With eplicit sequence formulas, when ou are looking for a specific term in the sequence, what do ou need to do? Sequences can also be described b using recursive definitions. When a sequence is defined recursivel, terms are found b operations on previous terms. A recursive definition alwas contains parts the first term and the formula. Eercise #: A sequence is defined b the recursive formula: f n f n 1 5 with f 1. Generate the first five terms of this sequence. Label each term with proper function notation. Eercise #: A sequence is defined recursivel as a1 ; an an 1 1. (a) What is the value of the second term in the sequence? (b) What is the value of the fourth term in the sequence? (c) When ou are looking for a specific term in a sequence defined recursivel, what must ou find first? Unit 5 Lesson 1 - Sequences Page 141

144 A l g e b r a U n i t 5 - Sequences and Series Eercise #4: Determine a recursive definition, in terms of include a starting value. Remember a recursive definition has parts. 5, 10, 0, 40, 80, 160, f n, for the sequence shown below. Be sure to Eercise #5: For the recursivel defined sequence (1) 18 () 456 () 8 (4) 1446 t n tn 1 and t 1, the value of t 4 is Eercise #6: Find an algebraic formula (eplicit), sequences. Recall that the domain that ou map from will be the set (a) 4, 5, 6, 7,... (b), 4, 8,16,... (c) an, similar to that in Eercise #1, for each of the following 1,,,..., n ,,,,... 4 d) ,,,, Eercise #7: Which of the following would represent the graph of the sequence an n 1? Eplain our choice. (1) () () (4) n n n n Eplanation: Unit 5 Lesson 1 - Sequences Page 14

145 A l g e b r a U n i t 5 - Sequences and Series Eercise #8: Match each of the eplicit and recursive formulas with its sequence of numbers. Eplicit Formula Recursive Formula Sequence 1. n1 A. f (1) 7, f ( n) f ( n 1) 6 W. -1,, 5, 8,. n 4 B. 1 a 6, a a 1 n n 1 X. 6,,, 4. 6n + 1 f (1), f ( n 1) f ( n) C. Y. 7, 1, 19, 5, 4. 1 n 1 D. a1 1, an an 1 Z., -6, 18, -54, Eercise #9: One of the most well-known sequences is the Fibonacci, which is defined recursivel using two previous terms. Its definition is: f n f n 1 f n and f 1 1 and f 1 Generate values for f, f 4, f 5, and f 6 (in other words, the net four terms of this sequence). Unit 5 Lesson 1 - Sequences Page 14

146 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 1 - SEQUENCES HOMEWORK 1. Given each of the following eplicit sequence definitions, write the first four terms. A variet of notations is used. (a) f n 7n f(1)= f()= f()= f(4)= (b) an n 5 (c) n t n (d) t n 1 n 1. Sequences below are defined recursivel. Determine and label the net three terms of the sequence. (a) f 1 4 and f n f n 1 8 (b) an an 1 a 1 and 1 4 (c) b b 1 n with 1 5 n n b (d) f n f n 1 n and f 1 4. Given the sequence 7, 11, 15, 19,..., which of the following represents an eplicit formula that will generate it? (1) an 4n 7 () an n 7 () an n 4 (4) an 4n 4. Which of the following formulas would represent the sequence 10, 0, 40, 80, 160, a () a (1) 10 n n () n 5 n a 10 n (4) a n 10 n n Unit 5 Lesson 1 - Sequences Page 144

147 A l g e b r a U n i t 5 - Sequences and Series HOMEWORK (cont.) 5. For each of the following sequences, determine an algebraic formula (eplicit), similar to Eercise #6, that defines the sequence. (a) 5, 10, 15, 0, (b), 9, 7, 81, (c) 1 4,,,, (d) 10, 0, 0, 40, (e) -, -4, -6, -8,. (f) 10, 100, 1000, 10000, 6. List the first 5 terms of the following recursive sequences: (a) a1 = an+1 = an (b) a1 = 81 an = 1 an-1 7. For each of the following sequences, state a recursive definition. Be sure to include a starting value. (a) 8, 6, 4,, (b), 6, 18, 54, (c),,,,... Unit 5 Lesson 1 - Sequences Page 145

148 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON -ARITHMETIC AND GEOMETRIC SEQUENCES In this lesson, we will review the basics of two particular sequences known as arithmetic (based on constant addition to get the net term) and geometric (based on constant multipling to get the net term). ARITHMETIC SEQUENCE EXPLICIT FORMULA where d is called the common difference and can be positive or negative. d = 1 Eercise #1: Generate the net three terms of the given arithmetic sequence an a n1 and a1 Eercise #: Find the first four terms of the given arithmetic sequence f n 6( n 1). Eercise #: Consider f ( n) f ( n 1) with f(1) 5. Is this a recursive or eplicit definition? th (a) Determine the value of f (), f (), and f (4). (b) Write an eplicit formula for the n term of an arithmetic sequence, f( n ), based on the first term, f (1), d and n. (c) Using our answer to (b), find f(1), f(), f(), and f(4) to make sure ou found the correct formula. Eercise #4: Given that a1 6 and a4 18 are members of an arithmetic sequence, determine the value of a 0. To find d, the common difference, think of average rate of change, use d = Unit 5 Lesson - Arithmetic and Geometric Sequences Page 146

149 A l g e b r a U n i t 5 - Sequences and Series Geometric sequences are defined ver similarl to arithmetic, but with a multiplicative constant instead of an additive one. GEOMETRIC SEQUENCE EXPLICIT FORMULA where r is called the common ratio and can be positive or negative and is sometimes fractional. or Eercise #5: Generate the net three terms of the geometric sequences given below. (b) f n f n 1 1 with (a) a1 4 and r f 1 9 (c) tn tn 1 with t1 Eercise #6:Determine if the sequence is arithmetic or geometric, write an eplicit formula and recursive formula. Unit 5 Lesson - Arithmetic and Geometric Sequences Page 147

150 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON - ARITHMETIC AND GEOMETRIC SEQUENCES HOMEWORK Use the given information to fill in the other three rows in the table. Hint: If the terms are not given, find the first few terms before completing the rest of the row. Terms Arithmetic or Geometric? Eplicit Formula Recursive Formula 1. 10, 14, 18,.... 0, 15, 7.5,.75, f (n) = f (n -1) with f(1) = 6 a n = a n-1-6 with a 1 = 0 5. f (n) = 5+ 1 (n -1) 6. a n = (-4) n-1 7. Consider f (n) = f (n -1) -10 with f(1) = 4. (a) Determine the value of f (), f (), and f (4). (b) Write an eplicit formula for the sequence, f( n ). th n term of the (c) Using our answer to (b), find f(1), f(), f(), and f(4) to make sure ou found the correct formula. Unit 5 Lesson - Arithmetic and Geometric Sequences Page 148

151 A l g e b r a U n i t 5 - Sequences and Series HOMEWORK (cont.) 8. Generate the net three terms of each arithmetic sequence shown below. (a) a1 and d 4 (b) a1, a 1 9. In an arithmetic sequence of numbers a1 4 and a6 46. Which of the following is the value of a 1? (1) 10 () 9 () 146 (4) The first term of an arithmetic sequence whose common difference is 7 and whose nd term is given b a 14 is which of the following? (1) 5 () 7 () 4 (4) Generate the net three terms of each geometric sequence defined below. (a) a1 8 with r 1 (b) a n an 1 and a In a geometric sequence, it is known that a1 1 and a4 64. The value of a 10 is (1) 65,56 () 51 () 6,144 (4) 4096 Unit 5 Lesson - Arithmetic and Geometric Sequences Page 149

152 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON - SUMMATION NOTATION Much of our work in this unit will concern adding the terms of a sequence. In order to specif this addition or summarize it, we introduce a new notation, known as summation or sigma notation that will represent these sums. This notation will also be used later in the course when we write formulas used in statistics. SUMMATION (SIGMA) NOTATION where i is called the inde variable, which starts at a value of a, ends at a value of n, and moves b unit increments (increase b 1 each time). Eercise #1: Evaluate each of the following sums. (a) 5 i (b) i 5 k 1 (c) j k1 j1 5 1 i 4 (e) 1( k 1) i1 k1 (d) (f) ii1 i1 Eercise #: Which of represents the value of (1) 1 10 () i1 1 i? () 9 4 (4) 1 4 Unit 5 Lesson - Summation Notation Page 150

153 A l g e b r a U n i t 5 - Sequences and Series Eercise #: Consider the sequence defined recursivel b an an 1 an and a1 0 and a 1. Find the value of 7 i4 a i Eercise #4: It is also good to be able to place sums into sigma notation. The values that are being summed in the net problems form either an arithmetic or geometric sequence. Look back at Eercise #1 on the previous page. Which problem represented the sum of the terms in an arithmetic sequence? A geometric sequence? Eercise #5: Epress each sum using sigma notation. Use n as our inde variable. First, consider an patterns ou notice amongst the terms involved in the sum. Then, work to put these patterns into a formula and sum. (a) (b) (c) (d) Eercise #6: Some sums are more interesting than others. Determine the value of reasoning. This is known as a telescoping series (or sum) Show our 1 i1 i i Unit 5 Lesson - Summation Notation Page 151

154 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON - SUMMATION NOTATION HOMEWORK 1. Evaluate each of the following. Place an non-integer answer in simplest rational form. (a) 5 4i (b) k 1 i k 0 (c) 4 n n1 n k (e) log10 i k 0 i1 (d) (f) k k. Which of the following is the value of 4k 1 (1) 5 () 7 () 45 (4) 80 4? k 0. The sum (1) i is equal to i4 () 4 () (4) Which of the following represents the sum ? 6 10 j 1 (1) 4j () j1 () j (4) j j 1 j j0 Unit 5 Lesson - Summation Notation Page 15

155 A l g e b r a U n i t 5 - Sequences and Series HOMEWORK (cont.) 5. Epress each sum using sigma notation. Use n as our inde variable. First, consider an patterns ou notice amongst the terms involved in the sum. Then, work to put these patterns into a formula and sum. (a) (b) (c) for 0 terms (d) A sequence is defined recursivel b the formula bn 4bn 1 bn with b1 1 and b. What is the value of 5 bi? Show the work that leads to our answer. i 7. A curious pattern occurs when we look at the behavior of the sum k 1 (a) Find the value of this sum for a variet of values of n below: k n 4 : k 1 n : 1 k 1 4 k 1 n. k 1 k n 5: k 1 n : 1 k 1 5 k 1 (b) What tpes of numbers are ou summing? What tpes of numbers are the sums? (c) Find the value of n such that. Unit 5 Lesson - Summation Notation Page 15

156 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 4 - ARITHMETIC SERIES A series is simpl the sum of the terms of a sequence. THE DEFINITION OF A SERIES If the set represent the elements of a sequence then the series,, is defined b: You have alread worked with series in previous lessons almost antime ou evaluated a summation problem. Eercise #1: Given the arithmetic sequence defined b a1 and an an 1 5, which of the following is the value of S 5 5 a? i1 i (1) () 5 () 40 (4) 7 The sums associated with arithmetic sequences, known as arithmetic series, have interesting properties, man applications and values that can be predicted with what is commonl known as rainbow addition. Eercise #: Consider the arithmetic sequence defined b a1 and an an 1. The series, based on the first eight terms of this sequence, is shown below. Terms have been paired off as shown. (a) What does each of the paired off sums equal? (b) How man of these pairs are there? (c) Using our answers to (a) and (b) find the value of the sum using a multiplicative process. (d) Generalize this now and create a formula for an arithmetic series sum based onl on its first term, a 1, its last term, a n, and the number of terms, n. Unit 5 Lesson 4 - Arithmetic Series Page 154

157 A l g e b r a U n i t 5 - Sequences and Series SUM OF AN ARITHMETIC SERIES Given an arithmetic series with n terms,, then its sum is given b: Eercise #: Which of the following is the sum of the first 100 natural numbers? Show the process that leads to our choice. (1) 5,000 () 10,000 () 5,100 (4) 5,050 Eercise #4: Find the sum of each arithmetic series described or shown below. (a) The sum of the siteen terms given b:. (b) The first term is, the common difference, d, is 6 and there are 0 terms (c) The last term is difference, d, is. and the common (d) The sum. Eercise #5: Kirk has set up a college savings account for his son, Mawell. If Kirk deposits $100 per month in an account, increasing the amount he deposits b $10 per month each month, then how much will be in the account after 10 ears? Unit 5 Lesson 4 - Arithmetic Series Page 155

158 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 4 - ARITHMETIC SERIES HOMEWORK 1. Which of the following represents the sum of if the arithmetic series has 14 terms? (1) 1,58 () 679 () 658 (4) 1,76. The sum of the first 50 natural numbers is (1) 1,75 () 1,50 () 1,875 (4) 950. If the first and last terms of an arithmetic series are 5 and 7, respectivel, and the series has a sum 19, then the number of terms in the series is (1) 18 () 14 () 11 (4) 1 4. Find the sum of each arithmetic series described or shown below. (a) The sum of the first 100 even, natural numbers. (b) The sum of multiples of five from 10 to 75, inclusive. (c) A series whose first two terms are and whose last term is 14. (d) A series of 0 terms whose last term is equal to 97 and whose common difference is five. Unit 5 Lesson 4 - Arithmetic Series Page 156

159 A l g e b r a U n i t 5 - Sequences and Series HOMEWORK (cont.) 5. For an arithmetic series that sums to 1,485, it is known that the first term equals 6 and the last term equals 9. Algebraicall determine the number of terms summed in this series. 6. Arlington High School recentl installed a new black-bo theatre for local productions. The onl had room for 14 rows of seats, where the number of seats in each row constitutes an arithmetic sequence starting with eight seats and increasing b two seats per row thereafter. How man seats are in the new black-bo theatre? Show the calculations that lead to our answer. 7. Simon starts a retirement account where he will place $50 into the account on the first month and increasing his deposit b $5 per month each month after. If he saves this wa for the net 0 ears, how much will the account contain? 8. The distance an object falls per second while onl under the influence of gravit forms an arithmetic sequence with it falling 16 feet in the first second, 48 feet in the second, 80 feet in the third, etcetera. What is the total distance an object will fall in 10 seconds? Show the work that leads to our answer. 9. A large grandfather clock strikes its bell once at 1:00, twice at :00, three times at :00, etcetera. What is the total number of times the bell will be struck in a da? Use an arithmetic series to help solve the problem and show how ou arrived at our answer. Unit 5 Lesson 4 - Arithmetic Series Page 157

160 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 5 - GEOMETRIC SERIES A series is simpl the sum of the terms of a sequence. THE DEFINITION OF A SERIES If the set represent the elements of a sequence then the series,, is defined b: In truth, ou have alread worked etensivel with series in previous lessons almost antime ou evaluated a summation problem. Eercise #1: Given a geometric series defined b the recursive formula a1 and an an 1, which of the following is the value of S 5 5 a? i1 (1) 106 () 9 () 75 (4) 5 i Eercise #: There is a formula for the sum of the first n terms of a geometric series, S n. The following is steps for deriving the formula. Steps 1. Write the eplicit formula for a geometric sequence ( a n form).. Write out n terms of the sequence b plugging in 1 through n.. Write an equation, S n, which gives the sum of these terms. 4. Multipl both sides of the equation b r. 5. Find, in simplest form, the value of S n - r S n (step minus step 4) 6. Write both sides of the equation in their factored form. 7. From the equation in step 6, find a formula for S in terms of a, r, and 1 n n b dividing b (1-r). a n = S n = S n r S n Work Unit 5 Lesson 5 - Geometric Series Page 158

161 A l g e b r a U n i t 5 - Sequences and Series For a geometric series defined b its first term, SUM OF A FINITE GEOMETRIC SERIES, and its common ratio, r, the sum of n terms is given b: or (don t memorize - it s on the formula sheet) Eercise #: Which of the following represents the sum of a geometric series with 8 terms whose first term is and whose common ratio is 4? (1),756 () 4,560 () 8,765 (4) 65,55 Eercise #4: Find the value of the geometric series shown below. Show the calculations that lead to our final answer Eercise #5: A person places 1 penn in a pigg bank on the first da of the month, pennies on the second da, 4 pennies on the third, and so on. Will this person be a millionaire at the end of a 1 da month? Eercise #6: You are offered a job that pas a salar of $51,000 the first ear and a % increase in each successive ear. You decide to accept the job. (a) What will our salar be during our tenth ear of emploment? (b) You worked ten ears for the compan. What are our total earnings? Eercise #7: Maria places $500 at the beginning of each ear into an account that earns 5% interest compounded annuall. Maria would like to determine how much mone is in her account after she has made her $500 deposit at the beginning of the 11 th ear (this amount would not get an interest). Unit 5 Lesson 5 - Geometric Series Page 159

162 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 5-GEOMETRIC SERIES HOMEWORK 1. Find the sums of geometric series with the following properties: (a) a1 6, r and n 8 (b) a 1 1 0, r, and n 6 (c) a 1 5, r, and n 10. If the geometric series (1) () has seven terms in its sum then the value of the sum is 7 () (4) Which of the following represents the value of the geometric series ? (1) 19,171 (),41 () 1,610 (4) 8, A geometric series has a first term of and a final term of and a common ratio of. The value of 4 this series is (1) ().5 () 16.5 (4) A geometric series whose first term is and whose common ratio is 4 sums to The number of terms in this sum is (1) 8 () 6 () 5 (4) 4 Unit 5 Lesson 5 - Geometric Series Page 160

163 A l g e b r a U n i t 5 - Sequences and Series HOMEWORK (cont.) 6. Find the sum of the geometric series shown below. Show the work that leads to our answer Ale earns $5,000 in his first ear of teaching and earns a % increase in each successive ear. Write a geometric series formula, Sn, for Ale s total earnings over n ears. Use this formula to find Ale s total earnings for his first 8 ears of teaching, to the nearest cent. 8. A college savings account is constructed so that $1000 is placed the account on Januar 1 st of each ear with a guaranteed % earl return in interest, applied at the end of each ear to the balance in the account. If this is repeatedl done, how much mone is in the account after the $1000 is deposited at the beginning of the 19 th ear? Show the sum that leads to our answer as well as relevant calculations. 9. A ball is dropped from 16 feet above a hard surface. After each time it hits the surface, it rebounds to a height that is of its previous maimum height. What is the total vertical distance, to the nearest foot, the ball has 4 traveled when it strikes the ground for the 10 th time? Write out the first five terms of this sum to help visualize. Unit 5 Lesson 5 - Geometric Series Page 161

164 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 6 - MORTGAGE PAYMENTS Mortgages are large amounts of mone borrowed from a bank. Monthl mortgage paments can be found using the formula below. This formula comes from a geometric series, but we will just be learning how to work with the formula and solve for the different variables. r r P M n r n M = monthl pament P = amount borrowed r = annual interest rate n = number of monthl paments The most basic wa to use the formula is to calculate monthl paments. 1. You took out a 0-ear mortgage for $0,000 to bu a house. The interest rate on the mortgage is 5.%. What are our monthl paments? When ou are taking out a mortgage, ou often know how much ou can afford each month, and ou want to determine what size mortgage ou can afford.. Based on our current income, ou can afford mortgage paments of $900 a month. You also want to take out a 15 ear mortgage to pa off the loan sooner. If the average interest rate at this time is.75%, what size mortgage can ou afford? The last wa we will learn to use the mortgage pament formula involves determining the length of the loan that ou can afford given the cost of a house, the amount ou can spend per month, and the interest rate.. Imagine ou have found the house of our dreams for $5,000. You know ou can afford monthl mortgage paments of $1500. You qualified for a mortgage with an interest rate of 4.75%. Algebraicall determine the number of paments ou would need to make to pa off the loan at this rate to the nearest whole number. How man ears would it take ou to the nearest tenth of a ear? Unit 5 Lesson 6 - Mortgage Paments Page 16

165 A l g e b r a U n i t 5 - Sequences and Series Eercise #1: And has a $00,000 mortgage at 4% earl interest. He s paing off his mortgage with $1,600 monthl paments (much of which are initiall going to interest). How much he still owes after n-paments is calculated with the following formula, where m is the monthl pament, r is the monthl interest rate, P is the principal, and n is the number of paments made. Find the amount owed on this loan after 1 ear and then after 10 ears. The amount owed after n paments, an, is: Eercise #: I paid $97,000 for m house in December of 008. The interest rate was 4.5% for 0 ears. Calculate the monthl pament (m) using the formula below. Recall that r is the monthl interest rate, P is the Pr principal, and n is the total number of paments. m n 1 1 r Eercise #: Using the formula from above: a) Determine the number of months it would take to pa off a $150,000 loan at a monthl 0.5% interest rate, with $1,000 paments. b) How much mone will it cost to pa off the loan when $1,000 is paed each month? Unit 5 Lesson 6 - Mortgage Paments Page 16

166 A l g e b r a U n i t 5 - Sequences and Series UNIT 5 LESSON 6 - MORTGAGE PAYMENTS HOMEWORK 1. You took out a 15-ear mortgage for $160,000 to bu a house. The interest rate on the mortgage is 5.%. a. What are our monthl paments?. Based on our current income, ou can afford mortgage paments of $150 a month. You also want to take out a 5-ear mortgage to spread the paments out over time. If the average interest rate at this time is.75%, what size mortgage can ou afford?. You have chosen a home in the perfect location for $50,000. You know ou can afford monthl mortgage paments of $1,400. You qualified for a mortgage with an interest rate of 4.75%. Algebraicall determine the number of paments ou would need to make to pa off the loan at this rate. How man ears would it take ou? Unit 5 Lesson 6 - Mortgage Paments Page 164

167 A l g e b r a U n i t 5 - Sequences and Series HOMEWORK (cont.) Use this formula for all problems on this page Pr m 1 1 n r 4. Calculate the monthl pament needed to pa off a $00,000 mortgage at 5.15% earl interest over 0 ears. Recall that r is the monthl rate. Show our work and carefull evaluate the above formula for m. 5. Do the same calculation as in the previous eercise but now make the pa off period 0 ears instead of 0. How much less is the monthl pament? 6. How man months would it take to pa off a $00,000 mortgage at a earl interest rate of 4.75% making monthl paments of $,500? 7. How much mone will it cost to pa off the loan if $,500 is paed each month? Unit 5 Lesson 6 - Mortgage Paments Page 165

168 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra U n i t 6 - Quadratic Functions and Their Algebra LESSON 1 FACTORING The definition of factor, in two forms, is given below. FACTOR TWO IMPORTANT MEANINGS (1) Factor (verb) To rewrite a quantit as an equivalent product. () Factor (noun) An individual component of a product. Alwas keep in mind that when we factor (verb) a quantit, we are rewriting it in a different form that is equal to the original quantit. Eercise #1: Rewrite each of the following binomials as a product of an integer and a binomial. a) 5 10 b) 6 c) 6 15 d) 6 14 The above tpe of factoring is called factoring out the greatest common factor (gcf). This greatest common factor can be numbers, variables, or both. Eercise #: Write each of the following binomials as the product of the binomial s gcf and another binomial. a) 6 b) 0 5 c) 10 5 d) 0 0 Eercise #: Rewrite each of the following trinomials as the product of its gcf and another trinomial. (a) 8 10 (b) (c) (d) The of a binomial is identical to it, but with the opposite sign in the middle. E. Consider the binomial + 5, this binomial has a conjugate: CONJUGATE MULTIPLICATION PATTERN If a binomial is of the special form a, known as the difference of perfect squares, then its factors are ( + a) and ( a). Eercise #4: Write each of the following binomials as the product of a conjugate pair. a) 9 b) c) 4 5 d) Eercise #5: Write each of the following binomials as the product of a conjugate pair. a) 1 b) b 6a c) d) 6 49 Unit 6 Lesson 1 - Factoring Page 166

169 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Factoring an epression until it cannot be factored an more is known as complete factoring. In general, when completel factoring an epression, the first tpe of factoring alwas to consider is factoring out the gcf. Eercise #6: Using a combination of gcf and difference of perfect squares factoring, write each of the following in its completel factored form. a) 5 0 b) 8 7 c) d) 48 Just as there is a pattern for factoring the difference of perfect squares, there are formulas for factoring the sum AND difference of perfect cubes. Eercise #7: To verif the sum of perfect cubes formula, simplif the following product: ( a b)( a ab b ) SUM OF PERFECT CUBES DIFFERENCE OF PERFECT CUBES Perfect Cubes: 1, 8,, 64,, Eercise #8: Factor each of the following epressions: a) + 8 b) c) 1000 d) - 15 e) f) 8 15 g) 50 4 h) 5 4² Unit 6 Lesson 1 - Factoring Page 167

170 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 1 FACTORING HOMEWORK 1. Rewrite each of the following binomials as the product of an integer and a different binomial. a) b) 4 40 c) 6 45 d) Rewrite each of the following binomials as the product of its gcf along with another binomial. a) b) 6 7 c) 0 5 d) Rewrite each of the following binomials as the product of a conjugate pair. a) 11 b) 64 c) 4 1 d) Rewrite each of the following trinomials as the product of its gcf and another trinomial. a) b) c) d) Completel factor each of the following binomials using a combination of gcf factoring and conjugate pairs. a) b) 6 4 c) 8 7 d) 7 1 e) f) 00 g) 8 51 h) Unit 6 Lesson 1 - Factoring Page 168

171 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) 1. The area of an rectangular shape is given b the product of its width and length. If the area of a particular rectangular garden is given b A 15 5 and its width is given b 5, then find an epression for the garden s length. Justif our response. 14. A projectile is fired from ground level such that its height, h, as a function of time, t, is given b h 16t 80t. Written in factored form this equation is equivalent to (1) h 16t t 4 () h 16t t 5 () h 8t t 7 (4) h 8t t 5 Unit 6 Lesson 1 - Factoring Page 169

172 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON - FACTORING BY GROUPING Toda we will introduce a new tpe of factoring known as factoring b grouping. This technique requires ou to see structure in epressions. Remember, whenever we factor we also look for a gcf FIRST! Warm up. Factor: p + 7p = Eercise #1: Factor a binomial common factor out of each of the following epressions. Write our final epression as the product of two binomials. a) b) 5 4 c) d) When a polnomial has 4 terms, ou will factor b grouping, follow the steps below. 1. Split the polnomial into the 1 st two terms, then the last two terms (draw line before the + or operator). Factor out the GCF of each set of two terms; what remains in both parentheses should be equal. Write what remains in parentheses, followed b the GCFs in their own set of parentheses Eamples: a) 4² + 8 b) ² c) 5 ² d) + 18² 18 Unit 6 Lesson - Factor B Grouping Page 170

173 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #: Use the method of factoring b grouping to completel factor the following epressions. (a) (b) (c) (d) (e) 7 18 (f) (g) (h) Eercise #: Consider the epression: Enter this epression in = on our calculator and find its zeroes (-intercepts). Use the following window: Xmin: -10, Xma:10, Ymin: -10, Yma: 50. Draw a rough sketch. Then, factor the epression completel. Do ou see the relationship between the factors and the zeroes? Unit 6 Lesson - Factor B Grouping Page 171

174 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON - FACTORING BY GROUPING HOMEWORK 1. Rewrite each of the following as the product of binomials. Be especiall careful on the manipulations that involve subtraction. a) 57 5 b) 4 c) 10 5 d) e) f) Ma tries to simplif the epression 5 as follows: Show using that this simplification is incorrect. Then, give the correct simplification.. Factor each of the following quadratic epressions completel using the method of grouping: a) b) Unit 6 Lesson - Factor B Grouping Page 17

175 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) 4. Factor each of the following cubic epressions completel. a) b) 18 7 c) 5 50 d) Factor each of the following epressions. Rearrange the epressions as needed to produce binomial pairs with common factors. a) ab a b b) ac c a Be careful when ou use factoring b grouping. Don't force the method when it does not appl. This can lead to errors. 6. Consider the epression Eplain the error made in factoring it. How can ou tell that the factoring is incorrect? Unit 6 Lesson - Factor B Grouping Page 17

176 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON FACTORING TRINOMIALS The abilit to factor trinomials, epressions of the form factored if the are the product of two binomials. Eercise #1: Warm up. Write each of the following products in simplest a b c, is an important skill. Trinomials can be a b c form. (a) 5 7 (b) 54 (c) 4 8 Factoring Trinomials tpes Factoring trinomials, epressions of the form a b c, when a = 1 - Alwas look for a gcf, first. - Find factors of the constant, c, that sum to the coefficient of the linear term, b. a) ² b) ² c) ² d) ² e) ² f) ² g) ² h) ² i) 10² j) 1 54 k) 5² l) ² Unit 6 Lesson - Factoring Trinomials Page 174

177 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Factoring trinomials, epressions of the form a b c, when a > 1 - Look for a gcf - Multipl the leading coefficient, a, and the constant, c. Write this product, a c. - Find the factors of a c that sum to the coefficient of the linear term, b. - Epress the linear term as a sum and proceed using factor b grouping. a) ² b) 4² c) 6² d) 6² e) ² f) 6² g) 6² 11 + h) ² i) ² Unit 6 Lesson - Factoring Trinomials Page 175

178 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON FACTORING TRINOMIALS HOMEWORK 1. Write each of the following trinomials in its factored form. (a = 1) Reminder, alwas look for gcf, first. a) 7 18 b) 14 4 c) 17 0 d) 5 6 e) 5 6 f) g) 1 0 h) 6 16 i) ² j) ² k) ² - 18 l) ² m) ² - 4 n) ² o) ² - 1 p) ² Write each of the following in its completel factored form. a) b) 6 4 c) 0 50 d) 75 Unit 6 Lesson - Factoring Trinomials Page 176

179 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.). Write each of the following trinomials in its factored form. (a > 1) a) b) 4 0 c) 9 15 d) e) f) 15² g) 9² h) ² 7 4 i) ² Completel factor each of the following. a) b) Unit 6 Lesson - Factoring Trinomials Page 177

180 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 4 FACTORING COMPLETELY Factoring can produce more than just two factors. In Eercise #1, we first warm-up b multipling three factors together. Eercise #1: Write each of these in their simplest form. The second question should take little time to do. (a) 4 7 (b) 4 To factor completel means ou have factored until ou can factor no more. Alwas look for a gcf, first. Factor completel a) ² b) 5³ 10² 40 c) ² 50 d) 18³ 98 e) ³ + 16² 10 f) 4 + 8³ 0² g) 6² h) 10³ 5² 5 i) 1u² 8u 4 Unit 6 Lesson 4 - Factoring completel Page 178

181 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra j) 7 5 4³ k) ³ ² l) 1³ + 6² + 7 m) n) 4 18 o) a ab a 6b p) q) 1 8 Unit 6 Lesson 4 - Factoring completel Page 179

182 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra 1. Factor completel: a) 14 6 UNIT 6 LESSON 4 FACTORING COMPLETELY HOMEWORK b) c) 19 d) e) 8 7 f) Write each of the following in completel factored form. a) b) More Practice Write each of the following epressions in its completel factored form. a) b) 45 0 Unit 6 Lesson 4 - Factoring completel Page 180

183 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) Note: there is one problem on this page of homework that is not factorable, which letter is it? c) d) e) 7 f) g) h) i) j) Unit 6 Lesson 4 - Factoring completel Page 181

184 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 5 - THE ZERO PRODUCT LAW One of the most important equation solving technique stems from a fact about the number zero that is not true of an other number: THE ZERO PRODUCT LAW If the product of multiple factors is equal to zero then at least one of the factors must be equal to zero. The law can immediatel be put to use in the first eercise. In this eercise, quadratic equations are given alread in factored form. Eercise #1: Solve each of the following equations for all value(s) of. a) 7 0 b) 54 0 c) Eercise #: In Eercise #1c), wh does the factor of 4 have no effect on the solution set of the equation? The Zero Product Law can be used to solve an quadratic equation that is factorable (not prime). To utilize this technique the problem solver must first set the equation equal to zero and then factor the non-zero side. Eercise #: Solve each of the following quadratic equations using the Zero Product Law. a) b) 1 7 Unit 6 Lesson 5 The Zero Product Law Page 18

185 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #4: Consider the sstem of equations shown below consisting of a parabola and a line. 8 5 and 4 5 a) Find the intersection points of these curves algebraicall. b) Using our calculator, sketch a graph of this sstem on the aes to the right. Be sure to label the curves with equations, the intersection points, and the window. c) Verif our answers to part a) b using the INTERSECT command on our calculator. The Zero Product Law is etremel important in finding the zero s or -intercepts (zeroes) of a parabola. Eercise #5: The parabola shown at the right has the equation. a) Write the coordinates of the two -intercepts of the graph. b) Find the -intercepts of this parabola algebraicall. Eercise #6: Algebraicall find the set of -intercepts (zeroes) for each parabola given below. a) 4 1 b) 1 10 c) 5 10 Unit 6 Lesson 5 The Zero Product Law Page 18

186 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 5 - THE ZERO PRODUCT LAW HOMEWORK 1. Solve each of the following equations for all value(s) of. a) 5 0 b) c) Solve each of the following quadratic equations which have alread been set equal to zero. a) b) c) Solve each of the following quadratic equations b first manipulating them so that one side of the equation is set equal to zero. a) b) 4 11 c) d) 16t 76t 5 1t 5 Unit 6 Lesson 5 The Zero Product Law Page 184

187 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) 4. Consider the sstem of equations shown below consisting of one linear and one quadratic equation. 4 5 and 5 10 a) Find the intersection points of this sstem algebraicall. b) Using our calculator, sketch a graph of this sstem to the right. Be sure to label the curves with equations, the intersection points, and the window. c) Use the INTERSECT command on our calculator to verif the results ou found in part a). 5. Algebraicall, find the zeroes (-intercepts) of each quadratic function given below. a) 81 b) 1 18 c) A quadratic function of the form b c. a) What are the -intercepts of this parabola? b) Based on our answer to part a), write the equation of this quadratic function first in factored form and then in trinomial form. Unit 6 Lesson 5 The Zero Product Law Page 185

188 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 6 - SOLVING INCOMPLETE QUADRATICS AND COMPLETING THE SQUARE Quadratics in the form a + c = 0are known as incomplete. Because these equations lack a linear (b) term the can be solved without the use of factoring and the Zero Product Law. Eercise #1: Solve each of the following incomplete quadratics for all values of. (a) -16 = 0 (b) 5-8 = 1 (c) = Unlike those quadratics that we factored and used the Zero Product Rule to solve, incomplete quadratics can have irrational answers as solutions. Eercise #: Solve each of the following incomplete quadratics for all values of. Place all answers in simplest radical form. (a) - 5 = 19 (b) 10 +1= 6 (c) 4 +5 = 8 (d) ( 5) = 54 An quadratic equation can be rewritten in a form where the method of Square roots can be used. This process is known as completing the square. Eample: Solve the equation, = Move the constant term to the other side of the equation. b b. Find and b. Complete the square b adding to both sides of the equation. 4. Factor the left side of the equation into a perfect square binomial. 5. Take the square root of both sides. Do not forget the plus or minus. 6. Add or subtract to solve for b b ( )( ) 17 ( ) You can use this process to solve an quadratic, whose lead coefficient, a, is 1, but it is especiall useful when the quadratic cannot be factored. Unit 6 Lesson 6 Complete the Square Page 186

189 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #: Solve each of the following prime quadratic equations b first completing the square. Epress our answers in simplest radical form. (a) = 0 (b) = 0 Eercise #4: For each of the following quadratics, epress our answers to the nearest hundredth. Graph the quadratic to verif that ou have found the correct answer for the zeroes. (a) + -1 = 0 (b) = 0 Quadratic equations where b is even and a=1 are the easiest to solve b completing the square. When b is odd, fractions are involved in the process. Eercise #5: Solve each of the following quadratic equations b completing the square. (a) 5 0 (b) 91 0 You cannot complete the square when a is greater than one. In those cases, divide the entire equation b a first and then complete the square. c) d) = 0 Unit 6 Lesson 6 Complete the Square Page 187

190 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 6 - SOLVING INCOMPLETE QUADRATICS AND COMPLETING THE SQUARE HOMEWORK 1. Solve each of the following incomplete quadratics. Epress our answers in simplest radical form when necessar. a) = 108 b) 1-7 = 5 c) = 0 d) 5 = 100 e) - 0 = 70 f) = 1. Solve each of the following quadratic equations b completing the square. Epress our answer in simplest radical form. a) - - = 0 b) = 0 c) = 0 d) -8 + = 0 Unit 6 Lesson 6 Complete the Square Page 188

191 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) e) = 0 f) = 0. Rounded to the nearest hundredth the larger root of is (1) 18.1 () 6.74 () 1.5 (4) Unit 6 Lesson 6 Complete the Square Page 189

192 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 7 - THE QUADRATIC FORMULA Simplif each of the following epressions: 11 6 = 50 6 = 7 6 = 5 6 = In the last lesson, ou solved the following quadratic equation, = 0 b completing the square. The solutions were = 5- and = 5+. Since an quadratic can be rearranged through the process of Completing the Square, a formula can be developed that will solve for the roots of an quadratic equation. This famous formula, known as the Quadratic Formula, is shown below. You worked with this as well in Algebra I. THE QUADRATIC FORMULA The solutions to the quadratic equation, assuming, are given b Eercise #1: Using the quadratic formula shown above, solve the equation = 0. You should get the same solution as ou did in the last lesson. How can ou tell from our solutions that the quadratic equation, = 0is not factorable? Eercise #: Which of the following represents the solutions to the equation ? (1) 5 10 () 5 10 () 5 5 (4) 5 5 Although the quadratic formula is most helpful when a quadratic epression is prime (not factorable), it can be used as a replacement for the Zero Product Law in cases where the quadratic can be factored. Unit 6 Lesson 7 The Quadratic Formula Page 190

193 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #: Solve the quadratic equation shown below in two different was (a) b factoring and (b) b using the quadratic formula. (a) b factoring (b) b the quadratic formula (c)where will the function, f () = intersect the -ais? The quadratic formula is ver useful in algebra - it should be committed to memor with practice. Eercise #4: Solve each of the following quadratic equations b using the quadratic formula. Some answers will be purel rational numbers and some will involve irrational numbers. Place all answers in simplest form. (a) 5 0 (b) 81 0 (c) 5 0 (d) Eercise #5: A shot-put throw can be modeled using the equation = where is the distance traveled (in feet) and is the height (also in feet). How long was the throw? Unit 6 Lesson 7 The Quadratic Formula Page 191

194 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 7 - THE QUADRATIC FORMULA HOMEWORK 1. Solve each of the following quadratic equations using the quadratic formula. Epress all answers in simplest form. (a) (b) 1 0 (c) 81 0 (d) 0 (e) (f) Unit 6 Lesson 7 The Quadratic Formula Page 19

195 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra. Which of the following represents all solutions of (1) 5 () 10 () 5 (4) 1 HOMEWORK (cont.) 41 0?. Which of the following is the solution set of the equation ? (1) 5 () 7 () (4) Rounded to the nearest hundredth the larger root of is (1) 18.1 () 6.74 () 1.5 (4) Algebraicall find the -intercepts of the quadratic function whose equation is Epress our answers in simplest radical form A missile is fired such that its height above the ground is given b h 9.8t 8.t 6.5, where t represents the number of seconds since the rocket was fired. Using the quadratic formula, determine, to the nearest tenth of a second, when the rocket will hit the ground. Unit 6 Lesson 7 The Quadratic Formula Page 19

196 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 8 - MORE WORK WITH QUADRATIC EQUATIONS Eercise #1: You have seen that some quadratics are factorable and some are not. You also know that certain methods can alwas be used to solve a quadratic. a) What are these methods? b) Let s sa ou used the quadratic formula to solve a quadratic. How can ou tell from our answers when ou could have also factored the quadratic? c) Decide if each set of numbers is rational, irrational, or nonreal. i. 7 7, ii. 5 5, iii. 0 0, iv. 9 9, d) When do ou usuall see numbers in the form above? e) What part of each number dictates what tpe of number it is? In the quadratic formula, b 4ac is the epression in the radical. It is known as the discriminant because helps ou discriminate (differentiate) between quadratics that can be factored and those that cannot be factored. (It also gives other information that will be covered later. Eercise #: Use the discriminant, b 4ac, to quickl determine if each of the following quadratics can be factored. Indicate if the quadratic has nonreal solutions. (a) ² + 4 = 0 (b) ² 7 6 = 0 (c) ² = 5 (d) ² 6 = 9 Eercise #: Which of the following sets represents the -intercepts of (1) () 1 7, , 6 6 () 5, 5 (4) 1,6 19 6? Unit 6 Lesson 8 More Work with Quadratic Equations Page 194

197 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #4: Consider the quadratic function f 4 6. (a) Algebraicall determine this function s -intercepts using the quadratic formula. Epress our answers in simplest radical form (b) Epress the -intercepts of the quadratic to the nearest hundredth. (c) Using our calculator, sketch a graph of the quadratic on the aes given. Use the intersect command on our calculator to verif our answers from part (b). (Remember to put =0) Label the zeros on the graph. Eercise #5: The Craz Carmel Corn compan modeled the percent of popcorn kernels that would pop, P, as a function of the oil temperature, T, in degrees Fahrenheit using the equation 1 P T.8T The compan would like to find the lowest temperature that ensures that at least 50% of the kernels will pop. Write an equation to model this situation. Solve this equation with the help of the quadratic formula. Round the temperature to the nearest tenth of a degree. Eercise #6: Find the intersection points of the linear-quadratic sstem shown below algebraicall. Then, use ou calculator to help produce a sketch of the sstem. Label the intersection points ou found on our graph. 4 6 and Unit 6 Lesson 8 More Work with Quadratic Equations Page 195

198 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 8 - MORE WORK WITH QUADRATIC EQUATIONS HOMEWORK 1. Use the discriminant, b 4ac, to quickl determine if each of the following quadratics can be factored. If the equation can be factored, solve b factoring. If the equation cannot be factored, choose a different method to solve it. (a) 5² 6 + = 0 (b) ² = 0. Which of the following represents the solutions to 4 1 6? (1) 4 7 () 5 () 5 11 (4) 4 1. The smaller root, to the nearest hundredth, of 1 0 is (1) 0.8 () 1.78 () 0.50 (4) The -intercepts of 7 0 are (1) () 5 6 and () and 5 (4) 11 Unit 6 Lesson 8 More Work with Quadratic Equations Page 196

199 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) 5. Solve the following equation for all values of. Epress our answers in simplest radical form Algebraicall solve the sstem of equations shown below and The Celsius temperature, C, of a chemical reaction increases and then decreases over time according to the 1 formula C t t 8t 9, where t represents the time in minutes. Use the Quadratic Formula to determine the least amount of time, to the nearest tenth of a minute, it takes for the reaction to reach 110 degrees Celsius. Unit 6 Lesson 8 More Work with Quadratic Equations Page 197

200 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 9 IMAGINARY NUMBERS Recall that in the Real Number Sstem, it is not possible to take the square root of a negative quantit because whenever a real number is squared it is non-negative. This fact has a ramification for finding the -intercepts of a parabola, as Eercise #1 will illustrate. Eercise #1: On the aes below, a sketch of f given in function notation as 1. is shown. Now, consider the parabola whose equation is (a) How is the graph of shifted to produce the graph of? (b) Create a quick sketch of on the aes below. (c) What can be said about the -intercepts of the function? (d) Algebraicall, show that these intercepts do not eist, in the Real Number Sstem, b solving the incomplete quadratic. Since we cannot solve this equation using Real Numbers, we introduce a new number, called i, the basis of imaginar numbers. Its definition allows us to now have a result when finding the square root of a negative real number. Its definition is given below. THE DEFINITION OF THE IMAGINARY NUMBER i Eercise #: Simplif each of the following square roots in terms of i. (a) 9 (b) 100 (c) (d) 18 Unit 6 Lesson 9 Imaginar Numbers Page 198

201 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #: Solve each of the following incomplete quadratics. Place our answers in simplest radical form. 1 (a) (b) 0 (c) 10 6 Eercise #4: Which of the following is equivalent to 5i 6i? (1) 0i () 0 () 11i (4) 11 Powers of i displa a remarkable pattern that allow us to simplif large powers of i into one of 4 cases. This pattern is discovered in Eercise #4. Eercise #5: Simplif each of the following powers of i. 1 i i i i 4 i 5 i 6 i 7 i 8 i We see, then, from this pattern that ever power of i is either 1,1, i, or Eercise #6: Simplif each of the following large powers of i. 1. Divide the large power b 4 noting the remainder. Write as i remainder and simplif i. And the pattern will repeat. (a) i 8 (b) i 1 (c) i 8 (d) i 40 Eercise #7: Which of the following is equivalent to i i i? (1) 8 i () 5 4i () 4 i (4) 7i Unit 6 Lesson 9 Imaginar Numbers Page 199

202 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra 1. The imaginar number i is defined as (1) 1 () 4 () 1 (4) 1 UNIT 6 LESSON 9 IMAGINARY NUMBERS HOMEWORK. Which of the following is equivalent to 18? (1) 8 () 8 () 8i (4) 8i. The sum 9 16 is equal to (1) 5 () 7i () 5i (4) 7 4. Which of the following powers of i is not equal to one? (1) i 16 () i () i 6 (4) i Which of the following represents all solutions to the equation ? (1) i () i () 5i (4) i 6. Solve each of the following incomplete quadratics. Epress our answers in simplest radical form. (a) (b) 0 Unit 6 Lesson 9 Imaginar Numbers Page 00

203 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) 7. Which of the following represents the solution set of (1) 7i () 5i () 7i (4) i 1 1 7? 8. Simplif each of the following powers of i into either 1,1, i, or i. (a) i (b) i (c) 4 i (d) i 11 (e) i 41 (f) i 0 (g) i 5 (h) i 6 (i) i 51 (j) i 45 (k) i 80 (l) i Which of the following is equivalent to i 7 i 8 i 9 i 10? (1) 1 () 1 i () i (4) When simplified the sum i 7i i 6i is equal to (1) 4i () 5 7i () i (4) 8 i 11. The product 6 i4 i can be written as (1) 4 6i () 5i () 18 10i (4) 0 10i Unit 6 Lesson 9 Imaginar Numbers Page 01

204 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 10 COMPLEX NUMBERS Comple numbers can alwas be thought of as a combination of a real number with an imaginar number and will have the form: a bi where a and b are real numbers We sa that a is the real part of the number and bi is the imaginar part of the number. These two parts, the real and imaginar, cannot be combined. Like real numbers, comple numbers ma be added and subtracted. The ke to these operations is that real components can combine with real components and imaginar with imaginar. Eercise #1: Find each of the following sums and differences. (a) 7i 6 i (b) 8 4i 1 i (c) 5i 7i (d) 5i 8 i Eercise #: Which of the following represents the sum of 6 i and 8 5i? (1) 5i () i () i (4) 5i Adding and subtracting comple numbers is straightforward because the process is similar to combining algebraic epressions that have like terms. Eercise #: Find the following products. Write each of our answers as a comple number in the form a bi. (a) 5i7 i (b) 6i i (c) 4i5 i Unit 6 Lesson 10 Comple Numbers Page 0

205 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra Eercise #4: Consider the more general product a bic di where constants a, b, c and d are real numbers. (a) Show that the real component of this product will alwas be ac bd. (b) Show that the product of i and 4 6i results in a purel real number. (c) Under what conditions will the product of two comple numbers alwas be a purel imaginar number? Check b generating a pair of comple numbers that have this tpe of product. Eercise #5: Determine the result of the calculation below in simplest a bi form. 5 i i 4i i Eercise #6: Which of the following products would be a purel real number? (1) 4i i () 5i5 i () i 4i (4) 6i6 i Unit 6 Lesson 10 Comple Numbers Page 0

206 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra 1. Find each of the following sum or difference. UNIT 6 LESSON 10 COMPLEX NUMBERS HOMEWORK (a) 6i 9i (b) 7 i 5i (c) 10 i 6 8i (d) 7i 15 6i (e) 15 i 5 5i (f) 1 i 5 6i. Which of the following is equivalent to 5 i 6i? (1) 9 18i () 9 6i () 1 8i (4) 1 i. Find each of the following products in simplest a bi form. (a) 5 i1 7i (b) 9i 4i (c) 4 i 6i 4. Comple conjugates are two comple numbers that have the form a bi and a bi. Find the following products of comple conjugates: (a) 57i5 7i (b) 10 i10 i (c) 8i 8i (d) What's true about the product of two comple conjugates? Unit 6 Lesson 10 Comple Numbers Page 04

207 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.) 5. Show that the product of a bi and a bi is the purel real number a b. 6. The product of 8 i and its conjugate is equal to (1) 64 4i () 68 () 60 (4) 60 4i 7. The comple computation 6 i6 i 4i 4i (1) 15 () 10 () 9 (4) 5 can be simplified to 8. Perform the following comple calculation. Epress our answer in simplest a bi form. 85i i 4i4 i 9. Perform the following comple calculation. Epress our answer in simplest a bi form. i i i Simplif the following comple epression. Write our answer in simplest a bi form. 5 i i Unit 6 Lesson 10 Comple Numbers Page 05

208 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 11 SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS As we saw in the last unit, the roots or zeroes of an quadratic equation can be found using the quadratic formula: b b 4ac Since this formula contains a square root, it is fair to investigate solutions to quadratic equations now when the quantit b 4ac, known as the discriminant, is negative. Up to this point, we would have concluded that if the discriminant was negative, the quadratic had no (real) solutions. But, now it can have comple solutions. Eercise #1: Use the quadratic formula to find all solutions to the following equation. Epress our answers in simplest a bi form. a As long as our solutions can include comple numbers, then an quadratic equation can be solved for two roots. Eercise #: Solve each of the following quadratic equations. Epress our answers in simplest a bi form. (a) (b) Unit 6 Lesson 11 Solving Quad. Equations w/comple solutions Page 06

209 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra There is an interesting connection between the -intercepts (zeroes) of a parabola and comple roots with nonzero imaginar parts. The net eercise illustrates this important concept. Eercise #: Consider the parabola whose equation is 6 1. (a) Algebraicall find the -intercepts of this parabola. Epress our answers in simplest a bi form. (b) Using our calculator, sketch a graph of the parabola on the aes below. Use the window indicated. 0 (c) From our answers to (a) and (b), what can be said about parabolas whose zeroes are comple roots with non-zero imaginar parts? 10 Eercise #4: Use the discriminant of each of the following quadratics to determine whether it has -intercepts. (a) 10 (b) 6 10 (c) 5 Eercise #5: Which of the following quadratic functions, when graphed, would not cross the -ais? (1) 5 () () 6 (4) 1 4 Unit 6 Lesson 11 Solving Quad. Equations w/comple solutions Page 07

210 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra UNIT 6 LESSON 11 SOLVING QUADRATIC EQUATIONS WITH COMPLEX SOLUTIONS HOMEWORK 1. Solve each of the following quadratic equations. Epress our solutions in simplest a bi form. Check. (a) (b) 1 (c) (d) (e) (f) Unit 6 Lesson 11 Solving Quad. Equations w/comple solutions Page 08

211 A l g e b r a U n i t 6 - Quadratic Functions and Their Algebra HOMEWORK (cont.). Which of the following represents the solution set to the equation (1) 1 or () i () 1 i (4) 1 i 0?. The solutions to the equation are (1) i () 6 i 11 () i (4) 6 i Using the discriminant, zeroes. b 4ac, determine whether each of the following quadratics has real or imaginar (a) 7 6 (b) 1 (c) 8 14 (d) 1 6 (e) 6 5 (f) Which of the following quadratics, if graphed, would lie entirel above the -ais? Tr to use the discriminant to solve this problem and then graph to check. (1) 1 () 4 7 () 6 (4) For what values of c will the quadratic for this problem. 6 c have no real zeroes? Set up and solve an inequalit Unit 6 Lesson 11 Solving Quad. Equations w/comple solutions Page 09

212 A l g e b r a U n i t 7 - Graphic Characteristics of Functions Even Functions U n i t 7 - Graphic Characteristics of Functions LESSON 1 - EVEN AND ODD FUNCTIONS A function is EVEN when its graph has smmetr about the -ais (like a reflection). The were named "even" functions because the functions, 4, 6, 8, etc. are smmetric across the -ais, but there are other functions that behave like that too, such as: For ever point (,) on the graph, (-, ) is also on the graph. An even eponent does not necessaril make an even function, for eample = (+4) is not an even function. Sketch it and show wh. Odd Functions A function is "odd" when it looks the same upside down, in other words, it is smmetric with respect to the origin, or has 180º rotational smmetr. For ever point (,) on the graph, (-, -) is also on the graph. The were named "odd" because the functions,, 5, 7, etc. have origin smmetr, but there are other functions that have 180º rotational smmetr: Unit 7 Lesson 1 Even and Odd Functions Page 10

213 A l g e b r a U n i t 7 - Graphic Characteristics of Functions An odd eponent does not necessaril make an odd function, for eample +1 is not an odd function. Sketch it and show wh. Neither Odd nor Even Don't be misled b the names "odd" and "even"... the are just names... and a function does not have to be even or odd. In fact most functions are neither odd nor even. For eample, Eample 1: Decide if the following graphs are even, odd, or neither. Eplain. (F) (G) (H) Unit 7 Lesson 1 Even and Odd Functions Page 11

214 A l g e b r a U n i t 7 - Graphic Characteristics of Functions. Graph f() = and tell if the function is odd, even, or neither.. Graph g() = and tell if the function is odd, even, or neither. 4. Graph h() = - 6 tell if the function is odd, even, or neither. Unit 7 Lesson 1 Even and Odd Functions Page 1

215 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON 1 - EVEN AND ODD FUNCTIONS HOMEWORK 1. Half of the graph of f is shown below. Sketch the other half based on the function tpe. (a) Even (b) Odd. Sketch the graph of the function and determine if the function is even, odd, or neither. a. f ( ) b. f ( ) 4 c. f ( ) 4 d. f ( ) e. g ( ) 1 f. g( ) 1 Unit 7 Lesson 1 Even and Odd Functions Page 1

216 A l g e b r a U n i t 7 - Graphic Characteristics of Functions g. HOMEWORK (cont.) g( ) h. f ( ) 1 j. f ( ) k. f ( ). If g is an odd, one-to-one function and if 1 graph of the inverse of g, g (1) 7, (), 7 (), 7 (4) 7, g 7, then which of the following points must lie on the. Eplain how ou made our choice. 4. Which of the following functions is even? (1) 4 () 9 () 6 (4) 4 5. The function f 4 is either even or odd. Determine which. 6. Even functions have smmetr across the -ais. Odd function have smmetr across the origin. Can a function have smmetr across the -ais? Wh or wh not? Unit 7 Lesson 1 Even and Odd Functions Page 14

217 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON - TRANSFORMATION OF FUNCTIONS Transformations of Graphs of Absolute Value and Quadratic Functions (use colored pencils) 1. a. Sketch the graph of b. Sketch the graph of 4 c. Sketch the graph of 5 d. Describe the graph of a in terms of the graph of. a. Sketch the graph of b. Sketch the graph of c. Sketch the graph of 6 d. Describe the graph of a in terms of the graph of. a. Sketch the graph of b. Sketch the graph of c. Describe the graph of in terms of the graph of Unit 7 Lesson Transformation of Functions Page 15

218 A l g e b r a U n i t 7 - Graphic Characteristics of Functions 4. a. Sketch the graph of b. Sketch the graph of 1 c. Sketch the graph of d. Describe the graph of a in terms of the graph of 5. a. Sketch the graph of b. Sketch the graph of 1 c. Sketch the graph of d. Describe the graph of (a) in terms of the graph of Recall transformations, such as line reflections, translations, and dilations. These transformations can be applied to basic functions. Parent Functions and Transformations There are certain basic functions whose graphs should be easil recognizable. Transformations will be applied to the graphs of these. These basic functions are also called parent functions. Below is a set of parent functions that will be used with transformations. f() = ² f() = ³ f() = f() = log() Unit 7 Lesson Transformation of Functions Page 16

219 A l g e b r a U n i t 7 - Graphic Characteristics of Functions Appling transformations to f() = log() Two different translations, or shifts, can be applied to a function a vertical shift or a horizontal shift. A vertical shift is when a constant is added/subtracted to f() = log() as follows: f() + = log() + f() 1 = log() 1 shifted up shifted down 1 A horizontal shift happens when a constant is added/subtracted to the inside parentheses: Note the difference from above! f(+) = log(+) shifted to the left f( ) = log( ) shifted to the right Two different reflections can be applied: in the -ais (negate the -values) or in the -ais (negate the -values). f() = log () f( ) = log ( ) Unit 7 Lesson Transformation of Functions Page 17

220 A l g e b r a U n i t 7 - Graphic Characteristics of Functions A vertical dilation occurs when the -values of each point are multiplied b a constant. f() = ² is a vertical STRETCH because a >1 ( taller ) (note: If a <1 it would be shorter ) Also known as a vertical compression A horizontal dilation occurs when the -values of each point are divided b a constant. 1 f = 1 is a horizontal STRETCH because a <1 ( fatter ) f = is a horizontal compression, SHRINK because a > 1 ( skinnier ) Unit 7 Lesson Transformation of Functions Page 18

221 A l g e b r a U n i t 7 - Graphic Characteristics of Functions Performing transformations with functions and relations Know: f( + c) f() + c f(-) -f() cf() f(c) Given the function f() state the transformation. a. f( ) b. f() + c. f() d. f(-) 1. Given the graph f() shown below, sketch the graphs of each and describe the transformation: a. f() + 5 b. f( 4) c. f() f() f() d. f(-) e. f() g. f(½) Function? One-to one? Original f() Domain and Range. Given the graph f() shown below, sketch the graphs of each and describe the transformation: a. f() + 6 b. f( + 5) c. f(-) d. f() f() f() e. ½ f() g. f( 1) + 5 Function? One-to one? Original f() Domain and Range Unit 7 Lesson Transformation of Functions Page 19

222 A l g e b r a U n i t 7 - Graphic Characteristics of Functions. Given the graph f() shown below, sketch the graphs of each and describe the transformation: a. f() 4 b. f( + 6) c. f(-) d. f() f() f() e. f() g. f( 7) + 8 Function? One-to one? Original f() Domain and Range 4. Given the graph f() = shown below, sketch the graphs of each and describe the transformation: a. f() 8 f() f() b. f( + 6) c. f(-) d. f() e. ½f() g. f( 1) + 6 Function? One-to one? Original f() Domain and Range Unit 7 Lesson Transformation of Functions Page 0

223 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON - TRANSFORMING FUNCTIONS HOMEWORK 1. Given the function f shown graphed on the grid, create a graph for each of the following functions and label on the grid. (a) g f (b) h f (c) k f 1 4. If the quadratic function b g f 4 5 f has a turning point at, 7 then where does the quadratic function g defined have a turning point? (1) 7,1 () 4, 5 () 1,1 (4) 4, 5. If the domain of f is 9 and the range of statements is correct about the domain and range of g f (1) Its domain is 1 11 and its range is 10. () Its domain is 5 7 and its range is 6 7. () Its domain is 1 11 and its range is 6 7. (4) Its domain is and its range is 10. f is 15, then which of the following 8? 4. Which of the following equations would represent the graph of the parabola reflection in the -ais? (1) 4 1 () after a () 4 1 (4) 4 1 Unit 7 Lesson Transformation of Functions Page 1

224 A l g e b r a U n i t 7 - Graphic Characteristics of Functions HOMEWORK (cont.) 5. The graph of 10 represents the graph of after (1) a vertical shift upwards of 10 units followed b a reflection in the -ais. () a reflection in the -ais followed b a vertical shift of 10 units upward. () a leftward shift of 10 units followed b a reflection in the -ais. (4) a reflection across the -ais followed b a rightward shift of 10 units. 6. If f 5 and of the following? g is the reflection of (1) g 5 () g 5 f across the -ais, then an equation of g is which () g 5 (4) 7. If the function f 4 graph of f? g 5 were graphed, it would represent which of the following transformations to the (1) A rightward shift of 4 units followed b a reflection in the -ais. () A rightward shift of 4 units followed b a reflection in the -ais. () A downward shift of 4 units followed b a reflection in the -ais. (4) A leftward shift of 4 units followed b a reflection in the -ais. 8. After a reflection in the -ais, the parabola 4 would have the equation (1) () 4 () 4 (4) If the point 6,10 lies on the graph of f of 1 f? (1), 5 () 6, 5 (),10 (4) 1, 0 then which of the following points must lie on the graph 10. If the function h is defined as vertical stretch b a factor of followed b a reflection in the -ais of the function f then h 1 f (1) f () 1 () f (4) f Unit 7 Lesson Transformation of Functions Page

225 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON GEOMETRIC DEFINITION OF A PARABOLA, PART 1 Materials: 1 sheet patt paper, 1 pencil, colored pencils, ruler Use ruler to draw a straight line one ruler width from the bottom of the patt paper Fold paper in half, folding line upon itself making a crease, mark a point above the line on this crease Make several creases in which the line coincides with the point Using a blue pencil outline the shape that the creases form A parabola is a special curve shaped like an arch. An point on a parabola is at an equal distance from.. *a fied point, the and * a fied straight line, The ais of smmetr is the line that divides a parabola into two parts that are mirror images. The verte of a parabola is the point at which the parabola intersects the ais of smmetr. Unit 7 Lesson Definition of a Parabola, part 1 Page

226 A l g e b r a U n i t 7 - Graphic Characteristics of Functions DEFINITION OF A PARABOLA A parabola is the set of all points (,) in a plane that are equidistant from a fied line (directri) and a fied point (focus) not on the line. The midpoint between the focus and the directri is called the verte, and the line passing through the focus and the verte is called the ais of smmetr. The directed distance from the focus to the verte is p. This new variable p is one ou'll need to be able to work with when writing equations of parabolas; it represents the distance between the verte and the focus, and also the distance between the verte and the directri so p is the distance between the focus and the directri. The standard form of the equation of a parabola with a vertical ais of smmetr when the verte (h,k) and the 1 p value are known is ( h) k 4 p When the ais of smmetr is a horizontal ais, the variables switch and we get: If lead coefficient is positive: open up, if negative: open down If lead coefficient is positive: open right, if negative: open left "regular", or vertical, parabola; the focus "inside" the parabola, the directri below the graph, the ais of smmetr passing through the focus and the verte "sidewas", or horizontal, parabola; the focus "inside" the parabola, the directri to the left of the graph, the ais of smmetr passing through the focus and the verte The important difference in the two equations is in which variable is squared: for regular (vertical) parabolas, the part is squared; for sidewas (horizontal) parabolas, the part is squared. For each of the following equations of a parabola (a) state whether it opens up or down, (b) verte, (c) p-value, (d) focus, and (e) find the equation of the directri ( ) 4 1 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Unit 7 Lesson Definition of a Parabola, part 1 Page 4

227 A l g e b r a U n i t 7 - Graphic Characteristics of Functions 1. 6 ( 1) 4 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri. 8( 8) ( 7) (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri 4. ( ) (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Unit 7 Lesson Definition of a Parabola, part 1 Page 5

228 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON DEFINITION OF A PARABOLA PART 1 HOMEWORK For each of the following equations of a parabola (a) state whether it opens up or down, (b) verte, (c) p-value, (d) focus, and (e) find the equation of directri ( ) 1 8 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri 1. 4 ( 4) 0 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri. 1( 6) ( 4) (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri Unit 7 Lesson Definition of a Parabola, part 1 Page 6

229 A l g e b r a U n i t 7 - Graphic Characteristics of Functions HOMEWORK (cont.) 4. ( 1) 4 (a) Opens up or down (b) Verte (c) p-value (d) Focus (e) Equation of directri 5. Write the equation of a vertical parabola whose verte is (-1,4) and p-value is. 6. Write the equation of a parabola whose verte is (-1,4) and focus is (-1,6). 7. Write the equation of a parabola whose verte is (,1) and focus is (,4). 8. Write the equation of a parabola whose verte is (,-) and focus is (,-5). Unit 7 Lesson Definition of a Parabola, part 1 Page 7

230 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON 4 GEOMETRIC DEFINITION OF A PARABOLA, PART Write the general equation of a parabola that has a vertical ais of smmetr. Write the general equation of a parabola that has a horizontal ais of smmetr. Eercises #1-4: For each of the following equations of a parabola (a) state whether it opens right, left, up or down, (b) verte, (c) p-value, (d) focus, and (e) find the equation of the directri ( ) 4 (a) Opens right or left (b) Verte (c) p-value (d) Focus (e) Equation of directri 1. 5 ( ) 16 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri Unit 7 Lesson 4 Definition of a Parabola, part Page 8

231 A l g e b r a U n i t 7 - Graphic Characteristics of Functions. 4( 6) ( ) (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 4. ( 1) 8 4 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 5. Write an equation for the set of points which are equidistant from the origin and the line = Write an equation for the set of points which are equidistant from (4,-) and the line = Write an equation for the set of points which are equidistant from (0,) and the -ais. Unit 7 Lesson 4 Definition of a Parabola, part Page 9

232 A l g e b r a U n i t 7 - Graphic Characteristics of Functions UNIT 7 LESSON 4 DEFINITION OF A PARABOLA PART HOMEWORK For each of the following equations of a parabola (a) state whether it opens right, left, up or down, (b) verte, (c) p-value, (d) focus, and (e) find the equation of directri ( ) 1 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri. 1 6 ( ) 0 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri. ( 1) 1 (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri Unit 7 Lesson 4 Definition of a Parabola, part Page 0

233 A l g e b r a U n i t 7 - Graphic Characteristics of Functions HOMEWORK (cont.) 4. ( ) 8( 5) (a) Opens (b) Verte (c) p-value (d) Focus (e) Equation of directri 5. Write the equation of the parabola with focus (1,6) and directri = Write the equation of the parabola with focus (-,0) and directri the -ais. 7. Write the equation of a parabola whose focus is (,1) and directri is = Write the equation of a parabola whose focus is (,) and directri is = - Unit 7 Lesson 4 Definition of a Parabola, part Page 1

234 A l g e b r a U n i t 7 - Graphic Characteristics of Functions LESSON 5 - CENTER-RADIUS EQUATION OF CIRCLES THE CENTER-RADIUS EQUATION OF A CIRCLE A circle whose center is at and whose radius is r is: Eercise #1: Which of the following equations would have a center of, 6 and a radius of? (1) 6 9 () 6 () 6 9 (4) 6 Eercise #: For each of the following equations of circles, determine both the circle s center and its radius. If its radius is not an integer, epress it in decimal form rounded to the nearest tenth. (a) (b) (c) 11 (d) 1 1 (e) 49 (f) (g) 64 (h) 4 0 (i) 57 Eercise #: Write equations for circles A and B shown below. Show how ou arrive at our answers. B A Unit 7 Lesson 5 Center-Radius Equation of Circles Page

235 A l g e b r a U n i t 7 - Graphic Characteristics of Functions Eercise #4 - Write each equation of a circle in center-radius form and identif the center and radius. a. ² + ² = 0 b. ² + ² = 0 c. 4² + 4² = 0 Sstems of Equations Involving a Circle and a Line Must solve for and - (,) Identif the ordered pairs which mark an intersection of a circle and a line. Here are three tpes of situations. Eercise #5: Solve the following sstem of equations algebraicall AND graphicall: 4 4 Unit 7 Lesson 5 Center-Radius Equation of Circles Page

236 A l g e b r a U n i t 7 - Graphic Characteristics of Functions LESSON 5 CENTER-RADIUS EQUATION OF CIRCLES HOMEWORK 1. Each of the following is an equation of a circle. State the circle s center and radius. In the cases where the radius is not an integer, give its value rounded to the nearest tenth. (a) 144 (b) ( ) ( 7) 6 (c) (d) (e) 1 (f) 5 5 (g) 50 (h) 00 (i) Which of the following is true about a circle whose equation is (1) It has a center of 5, and an area of 1. () It has a center of 5, and a diameter of 6. () It has a center of 5, and an area of 6. (4) It has a center of 5, and a circumference of ?. Which of the following represents the equation of the circle shown graphed below? (1) 16 () 4 () 4 (4) B completing the square on each of the quadratic epressions, determine the center and radius of a circle whose equation is shown below Unit 7 Lesson 5 Center-Radius Equation of Circles Page 4

237 A l g e b r a U n i t 7 - Graphic Characteristics of Functions HOMEWORK (cont.) 5. Circles are described below b the coordinates of their centers, C, and one point on their circumference, A. Determine an equation for each circle in center-radius form. 5, and 11,10 C, 5 and A, 17 C 5, 1 and A, 5 (a) C A (b) (c) 6. Solve the following sstem of equations graphicall Find the intersection of the circle 9 and algebraicall. 7. Jonas is designing a circular garden whose equation is 49. He wishes to place a walkwa within the garden at all points within the circle that satisf the inequalit. Graph the circle on the grid to the right and shade in all points that represent the walkwa. Unit 7 Lesson 5 Center-Radius Equation of Circles Page 5

238 A l g e b r a U n i t 8 - Etension Lessons for Honors course U n i t 8 - Etension Lessons for Honors course LESSON 1 LOG RULES Recall: Rules of Eponents Let a and b be positive integers and and real numbers: Multiplication Rule a b e. a 4 a = Division Rule a b 15 a e. 4 a Power Rule a b ( ) e. (a 4 ) = Recall: Log form of an eponential equation Eponential Form Logarithmic Form ³ = 8 log8 = 5 = 5 = Write each equation in logarithmic form Log Rules 1. Product Rule logb(ab) = or ln(ab) = A A. Quotient Rule logb = or ln = B B. Power Rule logb A or ln A *Note* There is no rule for logb(a+b) or logb(a B) These rules can be combined in a single epression. Also, alwas re-write radicals in eponential form e. = 1 5 Unit 8 Lesson 1 Log Rules Page 6

239 A l g e b r a U n i t 8 - Etension Lessons for Honors course Eamples: A. Epand each epression using the above properties: 1. log4(4). ln(³) 16. log a 4. ln(a² ) 5. log6 6. ln z 4 z 5 7. log ln e B. Condense each epression using the above properties: 1. log4(4) + log4(). ln() ln(). 1 log() + log() 4log(z) Unit 8 Lesson 1 Log Rules Page 7

240 A l g e b r a U n i t 8 - Etension Lessons for Honors course LESSON 1 LOG RULES HOMEWORK 4. ln(a) + 1 ln(b) ( ln(c) + ln(d) ) 5. log(51) log(64) 6. ln(e) + ln(e) + ln(e) 7. log(² 16) log( 4) Set A, do #s 4-5 Unit 8 Lesson 1 Log Rules Page 8

241 A l g e b r a U n i t 8 - Etension Lessons for Honors course HOMEWORK (cont.) Set B Epand and simplif when possible. 1. ln 4e. ln 4e. ln (4e) 4. ln e 1 5. ln e 6. ln e Condense log + 5log log z 8. ln - ln 9. ln e + ln e 10. log ½ (log + log z) Unit 8 Lesson 1 Log Rules Page 9

242 A l g e b r a U n i t 8 - Etension Lessons for Honors course LESSON PASCAL S TRIANGLE AND BINOMIAL EXPANSION From earlier work in algebra, ou are familiar with the term binomial, an epression consisting of two terms, such as +, or ² 5. These are binomials because the all show two monomial terms being combined b addition or subtraction. Binomial epansion is taking that two-term epression and raising it to successive powers, as shown below (a + b) 0 = 1 (a + b) 1 = 1a + 1b (a + b) = 1a² + ab + 1b² (a + b) = 1a + a²b 1 + a 1 b² + 1b (a + b) 4 = 1a 4 + 4a b 1 + 6a²b² + 4a 1 b + 1b 4 (a + b) 5 = 1a 5 + 5a 4 b a b² + 10a²b + 5a 1 b 4 + 1b 5 Notice that a binomial raised to the nth power has n + 1 terms. Look at the coefficients of each term in the epansion above and see if ou notice a pattern. This arrangement of numbers is known as Pascal s triangle after the French mathematician and phsicist Blaise Pascal. The following formula can be used to epand a binomial. ( + ) n = nco n 0 + nc1 n-1 1 +nc n- + + ncn-1 1 n-1 + ncn 0 n 1. Epand (m + ) Unit 8 Lesson Pascal s Triangle and Binomial Epansion Page 40

243 A l g e b r a U n i t 8 - Etension Lessons for Honors course. Epand (a ) 4. Epand ( + ) 5 4. Epand ( ) 6 Unit 8 Lesson Pascal s Triangle and Binomial Epansion Page 41

244 A l g e b r a U n i t 8 - Etension Lessons for Honors course LESSON PASCAL S TRIANGLE AND BINOMIAL EXPANSION HOMEWORK 1. Epand ( ) 5. Epand ( + 5) 4. Epand (a ) 5 Unit 8 Lesson Pascal s Triangle and Binomial Epansion Page 4

245 A l g e b r a U n i t 8 - Etension Lessons for Honors course 4. Epand (a 1) 6 HOMEWORK (cont.) 5. Epand ( + ) 5 6. Epand (a b) 6 Unit 8 Lesson Pascal s Triangle and Binomial Epansion Page 4

246 A l g e b r a U n i t 8 - Etension Lessons for Honors course LESSON BINOMIAL EXPANSION, CONT. Sometimes ou are asked for onl one term of a binomial epansion. You can epand the entire epression and select the particular term: 1. Find the third term of the epansion ( + ) 5. Find the middle term of the epansion (m 5) 6. Find the last term of the epansion ( + ) 5 Unit 8 Lesson Binomial Epansion, cont. Page 44

247 A l g e b r a U n i t 8 - Etension Lessons for Honors course 4. Find the middle term of the epansion (m 1) 8 5. Find the fourth term of the epansion ( + i) 5 6. Find the middle term of the epansion (m i) 6 Unit 8 Lesson Binomial Epansion, cont. Page 45

248 A l g e b r a U n i t 8 - Etension Lessons for Honors course LESSON BINOMIAL EXPANSION, CONT. HOMEWORK 1. Find the last term of the epansion ( - 4) 7. Find the third term of the epansion (i 6) 5. Find the middle term of the epansion ( - ) 6 Unit 8 Lesson Binomial Epansion, cont. Page 46

249 A l g e b r a U n i t 8 - Etension Lessons for Honors course HOMEWORK (cont.) 4. Find the third term of the epansion ( + 5) 5 5. Find the last term of the epansion (m i) 4 6. Find the fifth term of the epansion ( + ) 6 Unit 8 Lesson Binomial Epansion, cont. Page 47

250 A l g e b r a U n i t 9 - Regression U n i t 9 - Regression LESSON 1 - LINEAR REGRESSION AND PREDICTIONS A graph used to determine whether there is a relationship between paired data is called a. Recall how to view scatter plots and writing an equation of a line or curve of best fit using our graphing calculator: Calculator Steps: 1. Ke the sets of data into L 1 and L Stat Edit. Turn stat plot 1 on. nd =. Zoom 9 look at the scatter plot and note its shape. 4. Turn diagnostic on if not on alread. (CATALOG: nd 0 and scroll down) 5. Stat Calc (4:LinReg(a+b)) for a Linear Regression (if points are in a straight line) Under Store RegEQ, press VARS > Y-VARS > 1:Function > Y 1 Also recall: The number that is used to measure the strength and direction of a linear relationship is called the correlation coefficient, (denoted r). -1 r +1 If we have the Diagnostic On the calculator will compute the correlation coefficient (must go to CATALOG to turn Diagnostic On). A linear correlation coefficient close to zero signifies no significant linear correlation while a correlation coefficient close to +1 or -1 indicates that the points in the scatter plot are close to the calculated line of best fit, or are strongl correlated linearl. Linear Regression eample #1 Emma recentl purchased a new car. She decided to keep track of how man gallons of gas she used on five of her business trips. The results are shown in the table below. b) c) d) (a) Write a linear regression function g(m) for these data where miles driven, m, is the input, or independent variable. (Round all values to the nearest hundredth). (b) State the value of the correlation coefficient, r, to the nearest thousandth. (c) How man gallons would be used if 5 miles are driven (m=5)? (d) How far could Emma drive on her full tank of gas, if her tank holds 16 gallons? a) Unit 9 Lesson 1 - Linear Regression and Predictions Page 48

251 A l g e b r a U n i t 9 - Regression In the previous eample ou found a function value (using an input of m=5 miles) between given input values (150 m 1000). This process is called interpolation. Often we want to use data collected about past events to predict the future. The process of using given data values to approimate values outside the given range of values is called etrapolation. The same procedure is used, just find g(m) for an m value within or outside the given domain. Linear Regression eample #. The table below shows the attendance at a museum in select ears from 007 to 01. (a) State the linear regression function a(t) represented b the data table when t = 7 is used to represent the ear 007 and a(t) is used to represent the attendance. Round all values to the nearest hundredth. (b) State the correlation coefficient to the nearest hundredth and (c) determine whether the data suggest a strong or weak association. (d) What does the model predict the attendance to be this ear, in 016? (e) What ear would the attendance be 10.1 million? Linear Regression eample # Write a linear regression function f(t) for the following scatter plot: (Round to nearest hundredth) f(t) t Unit 9 Lesson 1 - Linear Regression and Predictions Page 49