Working with Functions

Working with Functions This V-shaped oga pose is called the "Boat Pose." Man people enjo the workout the get b doing oga. This pose reall works our abdominal muscles..1 This Will Be Absolutel FUN Graphing Absolute Value Functions............... 1005.2 Pla Ball! Absolute Value Equations and Inequalities.......... 1015.3 Make the Most of It Optimization................................. 1027.4 It s Not New, It s Reccled Composition of Functions...................... 1041.5 A Graph Is Worth a Thousand Words Interpreting Graphs............................ 1063 1003

1004

This Will Be Absolutel FUN Graphing Absolute Value Functions.1 LEARNING GOALS In this lesson ou will: Graph and analze the basic absolute value function. Investigate ke characteristics of the basic absolute value function. Graph transformations of absolute value functions. Analze transformations of absolute value functions using transformational function form. Generalize about the effects of transformations on ke characteristics of absolute value functions. Have someone in our class think of a whole number from 1 to 20. Ask each other student in the class to guess what the number is. Record all the guesses without revealing the mster number. On the graph shown, have the recorder determine each guess on the x-axis and plot its distance (shown on the -axis) from the mster number. What is the mster number? Did ou graph a function? Distance from Mster Number 20 19 18 17 16 15 14 12 11 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 14 15 16 17 18 19 20 Guess x 1005

PROBLEM 1 I ve Absolutel Done This Before 1. Graph the basic absolute value function f(x) 5 x. 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 2. Describe the domain and range of the basic absolute value function. Justif our reasoning. 3. How does the shape of the basic absolute value graph relate to its output values? 1006 Chapter Working with Functions

? 4. Justin makes a claim about the graph of the basic absolute value function. Justin The basic absolute value graph has an asmptote at 5 0, because the output values of an absolute value function are never negative. Is Justin s claim correct? Explain our reasoning. 5. Is an absolute value function invertible? Justif our reasoning. 6. Summarize the ke characteristics of the basic absolute value function. Basic Absolute Value Function Domain Range Asmptote Axis of Smmetr Vertex End Behavior Intervals of Increase or Decrease.1 Graphing Absolute Value Functions 1007

7. Graph each reflection of the basic absolute value function. a. a(x) 5 2 x 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 b. b(x) 5 2x 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 c. c(x) 5 2 2x 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 1008 Chapter Working with Functions

8. Describe the effect that a reflection over the line x 5 0 has on the domain and range of the basic absolute value function. Explain our reasoning. 9. Describe the effect that a reflection over the line 5 0 has on the domain and range of the basic absolute value function. Explain our reasoning..1 Graphing Absolute Value Functions 1009

PROBLEM 2 But Now I Feel Transformed! Recall that for each function famil, the basic function f(x) can be transformed b the function g(x) 5 Af(B(x 2 C)) 1 D. 1. For each transformation, describe the effect on the graph of the basic function. Then, write an example of an absolute value function for each transformation and describe its effect on the basic function. Function Form Description of Transformation of Graph Example 5 f(x) 1 D 5 f(x 2 C) 5 f(bx) 5 Af(x) 2. Graph each transformation of the basic absolute value function f(x) 5 x. Then, identif its vertex, and state its domain and range. a. g(x) 5 2f(x 1 2) Vertex: Domain: Range: 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 1010 Chapter Working with Functions

b. h(x) 5 1 4 f(x) 2 3 Vertex: Domain: Range: 8 6 4 2 28 26 24 22 22 24 26 28 0 2 4 6 8 x c. t(x) 5 f(3x) 1 1 Vertex: Domain: Range: 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 d. j(x) 5 f(23x) 1 1 Vertex: Domain: 8 6 Range: 4 2 28 26 24 22 22 24 0 2 4 6 8 x 26 28.1 Graphing Absolute Value Functions 1011

e. h(x) 5 f(2x 2 5) Vertex: Domain: Range: 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 3. What generalization can ou make about the effect of transformations on the domain of an absolute value function? 4. What generalization can ou make about the effect of transformations on the range of an absolute value function? 5. Use the transformation function g(x) 5 Af(B(x 2 C)) 1 D to write the general form for the coordinates of the vertex of an absolute value function. 1012 Chapter Working with Functions

6. Write a transformed absolute value function in terms of the basic absolute value function f(x) 5 x with the characteristics given. Then graph the transformed function. a. Translated up 6 units and verticall compressed b a factor of 1 3 units. 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 b. Vertex is at (3, 21) Range is (2`, 21] 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 c. Axis of smmetr is x 5 2 Horizontall stretched b a factor of 3 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28.1 Graphing Absolute Value Functions 10

d. Axis of smmetr is x 5 24 -intercept is (0, 2) 8 6 4 2 28 26 24 22 22 0 2 4 6 8 x 24 26 28 7. Summarize the ke characteristics of the transformed absolute value function g(x) 5 Af(B(x 2 C)) 1 D. Transformed Absolute Value Function Domain Range Asmptote Axis of Smmetr Vertex End Behavior Intervals of Increase or Decrease Be prepared to share our solutions and methods. 1014 Chapter Working with Functions

Pla Ball! Absolute Value Equations and Inequalities.2 LEARNING GOALS In this lesson, ou will: Understand and solve absolute values. Solve linear absolute value equations. Solve and graph linear absolute value inequalities on number lines. Graph linear absolute values and use the graph to determine solutions. KEY TERMS linear absolute value equation linear absolute value inequalit equivalent compound inequalit All games and sports have specific rules and regulations. There are rules about how man points each score is worth, what is in-bounds and what is out-ofbounds, and what is considered a penalt. These rules are usuall obvious to anone who watches a game. However, some of the regulations are not so obvious. For example, the National Hocke League created a rule that states that a blade of a hocke stick cannot be more than three inches or less than two inches in width at an point. In the National Football League, teams that wear black shoes must wear black shoelaces and teams that wear white shoes must wear white laces. In the National Basketball Association, the rim of the basket must be a circle exactl 18 inches in diameter. Most sports even have rules about how large the numbers on a plaer s jerse can be! Do ou think all these rules and regulations are important? Does it reall matter what color a plaer s shoelaces are? Wh do ou think professional sports have these rules, and how might the sport be different if these rules did not exist? 1015

PROBLEM 1 Too Heav? Too Light? You re Out! The official rules of baseball state that all baseballs used during professional games must be within a specified range of weights. The baseball manufacturer sets the target weight of the balls at 145.045 grams on its machines. The specified weight allows for a difference of 3.295 grams. This means the weight can be 3.295 grams greater than or less than the target weight. 1. Write an expression to represent the difference between a manufactured baseball s weight and the target weight. Use w to represent a manufactured baseball s weight. 2. Suppose the manufactured baseball has a weight that is greater than the target weight. a. Write an equation to represent the greatest acceptable difference in the weight of a baseball. b. Solve our equation to determine the greatest acceptable weight of a baseball. 3. Suppose the manufactured baseball has a weight that is less than the target weight. a. Write an equation to represent the least acceptable difference in weight. b. Solve our equation to determine the least acceptable weight of a baseball. The two equations ou wrote can be represented b the linear absolute value equation w 2 145.045 5 3.295. In order to solve an absolute value equation, recall the definition of absolute value. 1016 Chapter Working with Functions

4. Determine the solution(s) to each equation. 25 24 23 22 21 0 1 2 3 4 5 Use the number line as a tool to think about each solution. a. x 5 5 b. x 5 5 c. x 5 25 d. x 5 0 5. Analze each equation containing an absolute value smbol in Question 4. What does the form of the equation tell ou about the possible number of solutions?.2 Absolute Value Equations and Inequalities 1017

Consider this linear absolute value equation. a 5 6 There are two points that are 6 units awa from zero on the number line: one to the right of zero, and one to the left of zero. 1(a) 5 6 or 2(a) 5 6 Now consider the case where a 5 x 2 1. a 5 6 or a 5 26 x 2 1 5 6 If ou know that a 5 6 can be written as two separate equations, ou can rewrite an absolute value equation. 1(a) 5 6 or 2(a) 5 6 1(x 2 1) 5 6 or 2(x 2 1) 5 6 6. How do ou know the expressions 1(a) and 2(a) represent opposite distances? The expressions +(x 1) and (x 1) are opposites. 7. Determine the solution(s) to the linear absolute value equation x 2 1 5 6. Then check our answer. 1(x 2 1) 5 6 2(x 2 1) 5 6 To solve each equation, would it be more efficient to distribute the negative or divide both sides of the equation b 1 first? 1018 Chapter Working with Functions

8. Solve each linear absolute value equation. Show our work. a. x 1 7 5 3 b. x 2 9 5 12 Before ou start solving each equation, think about the number of solutions each equation ma have. You ma be able to save ourself some work and time!. c. 3x 1 7 5 28 d. 2x 1 3 5 0 9. Cho, Steve, Artie, and Donald each solved the equation x 2 4 5 5. Artie Donald x 4 = 5 (x) 4 = 5 (x) 4 = 5 (x) = 9 x = 9 x = 9 x 4 = 5 x = 9 (x) = 9 (x) = 9 x = 9 Cho x 4 = 5 (x) 4 = 5 [(x) 4] = 5 x 4 = 5 x + 4 = 5 x = 9 x = 1 x = 1 Steve x 4 = 5 (x) 4 = +5 (x) 4 = 5 x = 9 x 4 = 5 x = 1 x = 1.2 Absolute Value Equations and Inequalities 1019

a. Explain how Cho and Steve incorrectl rewrote the absolute value equation as two separate equations. b. Explain the difference in the strategies that Artie and Donald used. Which strateg do ou prefer? Wh? 10. Solve each linear absolute value equation. a. x 1 16 5 32 Consider isolating the absolute value part of the equation before ou rewrite as two equations. b. 23 5 x 2 8 1 6 c. 3 x 2 2 5 12 d. 35 5 5 x 1 6 2 10 1020 Chapter Working with Functions

11. The average width of a field hocke stick is 48 mm. It can var b 2.75 mm and still meet regulations for varsit pla. a. Formulate an absolute value linear equation to represent this situation. b. Solve the absolute value linear equation to determine the maximum and minimum allowable width for a varsit field hocke plaer s stick. PROBLEM 2 Too Big? Too Small? Just Right. In Too Heav? Too Light? You re Out! ou determined the linear absolute value equation to identif the most and least a baseball could weigh and still be within the specifications. The manufacturer wants to determine all of the acceptable weights that the baseball could be and still fit within the specifications. You can write a linear absolute value inequalit to represent this problem situation. 1. Write a linear absolute value inequalit to represent all baseball weights that are within the specifications. 2. Determine if each baseball has an acceptable weight. Explain our reasoning. a. A manufactured baseball weighs 147 grams..2 Absolute Value Equations and Inequalities 1021

b. A manufactured baseball weighs 140.8 grams. c. A manufactured baseball weighs 148.34 grams. d. A manufactured baseball weighs 141.75 grams. 3. Complete the inequalit to describe all the acceptable weights, where w is the baseball s weight. Then use the number line to graph this inequalit. # w # 140 141 142 143 144 145 146 147 148 149 150 1022 Chapter Working with Functions

4. Ramond has the job of disposing of all baseballs that are not within the acceptable weight limits. a. Write an inequalit to represent the weights of baseballs that Ramond can dispose of. b. Graph the inequalit on the number line. 140 141 142 143 144 145 146 147 148 149 150 In Little League Baseball, the diameter of the ball is slightl smaller than that of a professional baseball. 5. The same manufacturer also makes Little League baseballs. For these baseballs, the manufacturer sets the target diameter to be 7.47 centimeters. The specified diameter allows for a difference of 1.27 centimeters. a. Denise measures the diameter of the Little League baseballs as the are being made. Complete the table to determine each difference. Then write the linear absolute value expression used to determine the diameter differences. Independent Quantit Diameter of the Little League Baseballs Dependent Quantit Target and Actual Diameter Difference Units 6.54 8.75 7.39 5.99 8 9.34 7.47 d.2 Absolute Value Equations and Inequalities 1023

b. Graph the linear absolute value function, f(d ), on a graphing calculator. Sketch the graph on the coordinate plane. Be sure to label our axes. 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0 5 5.5 6 6.5 7 7.5 8 8.5 9 x 6. Determine the diameters of all Little League baseballs that fit within the specifications. a. Use our graph to estimate the diameters of all the Little League baseballs that fit within the specifications. Explain how ou determined our answer. b. Algebraicall determine the diameters of all the baseballs that fit within the specification. Write our answer as an inequalit. 1024 Chapter Working with Functions

7. The manufacturer knows that the closer the diameter of the baseball is to the target, the more likel it is to be sold. The manufacturer decides to onl keep the baseballs that are less than 0.75 centimeter from the target diameter. a. Algebraicall determine which baseballs will not fall within the new specified limits and will not be kept. Write our answer as an inequalit. b. How can ou use our graph to determine if ou are correct? Talk the Talk Absolute value inequalities can take four different forms as shown in the table. To solve a linear absolute value inequalit, ou must first write it as an equivalent compound inequalit. Notice that the equivalent compound inequalities do not contain absolute values. Absolute Value Inequalit ax 1 b, c Equivalent Compound Inequalit 2c, ax 1 b, c ax 1 b # c 2c # ax 1 b # c ax 1 b. c ax 1 b, 2c or ax 1 b. c ax 1 b $ c ax 1 b # 2c or ax 1 b $ c 1. Solve the linear absolute value inequalit b rewriting it as an equivalent compound inequalit. Then graph our solution on the number line. a. x 1 3, 4 210 28 26 24 22 0 2 4 6 8 10 As a final step, don t forget to check our solution..2 Absolute Value Equations and Inequalities 1025

b. 6 # 2x 2 4 210 28 26 24 22 0 2 4 6 8 10 c. 25x 1 8 1 2, 25 210 28 26 24 22 0 2 4 6 8 10 d. x 1 5. 21 210 28 26 24 22 0 2 4 6 8 10 e. x 1 5, 21 210 28 26 24 22 0 2 4 6 8 10 Be prepared to share our solutions and methods. 1026 Chapter Working with Functions

Make the Most of It Optimization.3 LEARNING GOALS In this lesson, ou will: Formulate sstems of linear inequalities in two variables. Solve sstems of two or more linear inequalities in two variables. Determine possible solutions in the solution set of sstems of two or more linear inequalities in two variables. Determine constraints from a problem situation. Analze a function to calculate maximum or minimum values. KEY TERMS solution of a sstem of inequalities linear programming Optimization problems involve finding the best solution from a choice of solutions given the objective. One of the most famous optimization problems in mathematics is the Traveling Salesman Problem. This problem can be stated as follows: Given a number of cities connected b roads, describe the shortest route that can be taken b a traveling salesman in order to visit ever cit once and then return to his starting place. Work on problems like these has potential applications in computer science and other fields. But a solution ma forever remain elusive. In 1972, a computer scientist named Richard Karp showed that a solution to the Traveling Salesman problem might not even be possible! 1027

PROBLEM 1 Beep. Boop. Time Is Mone 1. Janice works two jobs and is tring to maximize her weekl earnings. Her bab-sitting job pas $10 per hour, and her job at an ice cream store pas $7.50 per hour. Suppose she needs to earn at least $90 to cover her weekl expenses. a. Write a linear inequalit to represent the total amount of mone Janice needs to earn at her jobs. Be sure to define our variables. b. Graph the linear inequalit on the coordinate plane. Hours at Ice Cream Store 30 20 10 10 20 30 40x Hours Bab-Sitting c. Janice s friend Colton suggests that she work 18 hours at her bab-sitting job and 12 hours at the ice cream shop. If Janice follows Colton s advice, will she earn enough to cover her weekl expenses? Explain our reasoning. 1028 Chapter Working with Functions

d. Write this solution as an ordered pair. Where does it fall on the graph? e. Determine another possible solution to this problem situation. Justif our reasoning. f. Wh is the graph restricted to the first quadrant? g. Write two inequalities that represent this restriction. 2. Suppose that Janice can onl work a maximum of 20 total hours per week. a. Write a linear inequalit to represent the total number of hours that Janice could work per week. b. Is Colton s suggestion of 18 hours at her bab-sitting job and 12 hours at the ice cream shop still reasonable? Justif our reasoning. c. Formulate the four linear inequalities that ou have written into a sstem of inequalities that represents Janice s problem situation. It is important to alwas consider which values are reasonable for each variable in the problem situation. In Janice s case, because there is no such thing as negative hours, the variables must be restricted to non-negative numbers..3 Optimization 1029

d. Graph the sstem of linear inequalities on one coordinate plane. Hours at Ice Cream Store 30 20 10 10 20 30 40x Hours Bab-Sitting e. Suppose that Janice is onl able to bab-sit for 6 hours. How man hours must she work at the ice cream shop in order to meet her earnings requirement? Justif our reasoning. f. Write our solution as an ordered pair, and plot it on the graph in part (d). Where does it fall? g. Erin suggests that Janice also work 10 hours at the ice cream shop. Is this solution reasonable? How does it compare to our answer in part (e)? 1030 Chapter Working with Functions

h. Write Erin s solution as an ordered pair and plot it on the graph in part (d). Where does it fall? i. Compare these solutions with the ordered pair (18, 12) suggested b Colton in part (b). What do ou notice? The solution of a sstem of inequalities is the intersection of the solutions to each inequalit. Ever point in the intersection satisfies all inequalities in the sstem. 3. Select one ordered pair that represents a point inside the overlapping region. Verif that it is a solution to the sstem of linear inequalities. 4. Select one ordered pair that represents a point outside the overlapping region. Verif that it is not a solution to the sstem of linear inequalities..3 Optimization 1031

5. Serena and Adam were each asked to graph the sstem of linear inequalities shown..23x 1 5,23x 2 2?Serena Adam 8 8 6 6 4 4 2 2 28 26 24 22 22 0 2 4 6 8 x 28 26 24 22 22 0 2 4 6 8 x 24 24 26 26 28 28 The two inequalities do not overlap, and therefore the sstem has no solution. The solution to the sstem of inequalities falls in between the two lines. a. Who is correct? Justif our reasoning. b. Select an ordered pair from each region of the graph to verif our response. 1032 Chapter Working with Functions

6. Wh are the lines in Question 5 dashed? 7. Will two parallel lines alwas produce a sstem of linear inequalities with no solution? 8. The Whistling Baker is a baker that specializes in cupcakes and muffins. The sell cupcakes for $2.00 each and muffins for $1.25 each. The Whistling Baker can bake a total of 8 dozen treats each da, and needs to make $100 per da. a. Formulate a sstem of linear inequalities to represent this problem situation. Be sure to define our variables..3 Optimization 1033

b. Graph the sstem of linear inequalities on the coordinate plane. 90 80 Number of Muffins Sold 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 Number of Cupcakes Sold x c. Identif one possible solution. Is this reasonable within the context of the problem situation? 1034 Chapter Working with Functions

PROBLEM 2 Getting in Shape Brad is a distance runner who wants to design the most effective workout plan to prepare for his next race. To adequatel prepare for the race, Brad must run between 3 and 6 hours per week, but he knows that he must also spend some time strength training to reduce the risk of injur. Brad wants to devote at least twice as much time to running as to strength training, but he can spend no more than 8 hours working out each week. 1. Write a sstem of inequalities to represent the constraints of this problem situation. Be sure to define our unknowns. 2. Graph the sstem of inequalities on the coordinate plane shown. Shade the region that represents the solution set. 9 Time Spent Strength Training (hours) 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 9 x Time Spent Running (hours).3 Optimization 1035

You can use the intersection points to find the optimal solution to a sstem of inequalities through a process called linear programming. In linear programming, the vertices of the solution region of the sstem of linear inequalities are substituted in to a given equation to find the maximum or minimum value. 3. Label the vertices of the shaded region. 4. Running burns 600 calories per hour, while strength training burns 250 calories per hour. Write an equation to represent this relationship. Be sure to define our variables. 5. Use the vertices of the solution region to determine how man hours Brad should devote to each tpe of exercise in order to maximize his weekl workout. 6. One of Brad s friends was interested in losing a little weight. He follows Brad s training program to tr to lose one pound per week. In order to lose a pound a week, he needs to burn a total of 3500 calories each week. Will Brad s friend meet his goal of losing one pound per week? Explain our reasoning. 1036 Chapter Working with Functions

PROBLEM 3 It s Prom Time! Last ear, tickets to the prom cost $50 per student, and 120 students attended. A student surve found that for ever $5 reduction in price, 30 more students would attend. 1. Write a function to model the revenue generated from the sale of prom tickets for ever x reduction in price. 2. Calculate the x- and -intercept and explain what each means within the context of this problem situation..3 Optimization 1037

3. Determine the ticket price that will maximize revenue. What is the shape of the graph? How can I determine the maximum? 4. Calculate the amount of revenue that the new ticket price will generate. 1038 Chapter Working with Functions

5. The venue for the prom charges the school a flat rate of $2000 for up to 150 students, and then $15 for ever additional student. a. Calculate the profit that the prom made for the school last ear. b. Determine the profit that the school will make if the use the new ticket price. 6. Compare the profit that the prom made for the school last ear with the profit that it will make this ear if it uses the new ticket price. Is it worth it to change the ticket price? Explain our reasoning. Be prepared to share our solutions and methods..3 Optimization 1039

1040 Chapter Working with Functions