Graph Quadratic Functions in Standard Form

TEKS 4. 2A.4.A, 2A.4.B, 2A.6.B, 2A.8.A Graph Quadratic Functions in Standard Form Before You graphed linear functions. Now You will graph quadratic functions. Wh? So ou can model sports revenue, as in Eample 5. Ke Vocabular quadratic function parabola verte ais of smmetr minimum value maimum value A quadratic function is a function that can be written in the standard form 5 a 2 b c where a Þ 0. The graph of a quadratic function is a parabola. KEY CONCEPT Parent Function for Quadratic Functions For Your Notebook The parent function for the famil of all quadratic functions is f() 5 2. The graph of f() 5 2 is the parabola shown below. The lowest or highest point on a parabola is the verte. The verte for f() 5 2 is (0, 0). 5 2 The ais of smmetr divides the parabola into mirror images and passes through the verte. For f() 5 2, and for an quadratic function g() 5 a 2 b c where b 5 0, the verte lies on the -ais and the ais of smmetr is 5 0. E XAMPLE Graph a function of the form 5 a 2 Graph 5 2 2. Compare the graph with the graph of 5 2. STEP Make a table of values for 5 2 2. SKETCH A GRAPH Choose values of on both sides of the ais of smmetr 5 0. STEP 2 STEP 3 22 2 0 2 8 2 0 2 8 Plot the points from the table. Draw a smooth curve through the points. 3 5 2 5 2 2 STEP 4 Compare the graphs of 5 2 2 and 5 2. Both open up and have the same verte and ais of smmetr. The graph of 5 2 2 is narrower than the graph of 5 2. 236 Chapter 4 Quadratic Functions and Factoring

E XAMPLE 2 Graph a function of the form 5 a 2 c Graph 52 } 2 2 3. Compare the graph with the graph of 5 2. STEP Make a table of values for 52} 2 3. 2 SKETCH A GRAPH Choose values of that are multiples of 2 so that the values of will be integers. STEP 2 STEP 3 STEP 4 24 22 0 2 4 5 2 25 3 25 Plot the points from the table. Draw a smooth curve through the points. 52 2 3 2 Compare the graphs of 52} 2 3 and 2 5 2. Both graphs have the same ais of smmetr. However, the graph of 52} 2 3 opens down and is 2 wider than the graph of 5 2. Also, its verte is 3 units higher. GUIDED PRACTICE for Eamples and 2 Graph the function. Compare the graph with the graph of 5 2.. 524 2 2. 52 2 2 5 3. f() 5 } 4 2 2 GRAPHING ANY QUADRATIC FUNCTION You can use the following properties to graph an quadratic function 5 a 2 b c, including a function where b Þ 0. KEY CONCEPT For Your Notebook Properties of the Graph of 5 a 2 b c 5 a 2 b c, a > 0 5 a 2 b c, a < 0 52 b 2a (0, c) 52 b 2a (0, c) Characteristics of the graph of 5 a 2 b c: The graph opens up if a > 0 and opens down if a < 0. The graph is narrower than the graph of 5 2 if a > and wider if a <. The ais of smmetr is 52} b and the verte has -coordinate 2} b. 2a 2a The -intercept is c. So, the point (0, c) is on the parabola. 4. Graph Quadratic Functions in Standard Form 237

E XAMPLE 3 Graph a function of the form 5 a 2 b c Graph 5 2 2 2 8 6. AVOID ERRORS Be sure to include the negative sign before the fraction when calculating the -coordinate of the verte. STEP Identif the coefficients of the function. The coefficients are a 5 2, b 528, and c 5 6. Because a > 0, the parabola opens up. STEP 2 Find the verte. Calculate the -coordinate. 52} b 52 } (28) 5 2 2a 2(2) Then find the -coordinate of the verte. 5 2(2) 2 2 8(2) 6 522 So, the verte is (2, 22). Plot this point. STEP 3 Draw the ais of smmetr 5 2. STEP 4 Identif the -intercept c, which is 6. Plot the point (0, 6). Then reflect this point in the ais of smmetr to plot another point, (4, 6). STEP 5 Evaluate the function for another value of, such as 5. ais of smmetr 5 2 verte (2, 22) 5 2() 2 2 8() 6 5 0 STEP 6 Plot the point (, 0) and its reflection (3, 0) in the ais of smmetr. Draw a parabola through the plotted points. (2, 22) 5 at classzone.com GUIDED PRACTICE for Eample 3 Graph the function. Label the verte and ais of smmetr. 4. 5 2 2 2 2 5. 5 2 2 6 3 6. f() 52 } 3 2 2 5 2 KEY CONCEPT For Your Notebook Minimum and Maimum Values Words For 5 a 2 b c, the verte s -coordinate is the minimum value of the function if a > 0 and the maimum value if a < 0. Graphs maimum minimum a is positive a is negative 238 Chapter 4 Quadratic Functions and Factoring

E XAMPLE 4 Find the minimum or maimum value Tell whether the function 5 3 2 2 8 20 has a minimum value or a maimum value. Then find the minimum or maimum value. Because a > 0, the function has a minimum value. To find it, calculate the coordinates of the verte. 52} b 52 } (28) 5 3 2a 2(3) 5 3(3) 2 2 8(3) 20 527 c The minimum value is 527. You can check the answer on a graphing calculator. Minimum X=3 Y=-7 E XAMPLE 5 TAKS REASONING: Multi-Step Problem GO-CARTS A go-cart track has about 380 racers per week and charges each racer $35 to race. The owner estimates that there will be 20 more racers per week for ever $ reduction in the price per racer. How can the owner of the go-cart track maimize weekl revenue? STEP STEP 2 Define the variables. Let represent the price reduction and R() represent the weekl revenue. Write a verbal model. Then write and simplif a quadratic function. Revenue (dollars) 5 Price (dollars/racer) p Attendance (racers) INTERPRET FUNCTIONS Notice that a 5220 < 0, so the revenue function has a maimum value. STEP 3 R() 5 (35 2 ) p (380 20) R() 5 3,300 700 2 380 2 20 2 R() 5 220 2 320 3,300 Find the coordinates (, R()) of the verte. 52 b } 2a 52 320 } 2(220) 5 8 Find -coordinate. R(8) 5220(8) 2 320(8) 3,300 5 4,580 Evaluate R(8). c The verte is (8, 4,580), which means the owner should reduce the price per racer b $8 to increase the weekl revenue to $4,580. GUIDED PRACTICE for Eamples 4 and 5 7. Find the minimum value of 5 4 2 6 2 3. 8. WHAT IF? In Eample 5, suppose each $ reduction in the price per racer brings in 40 more racers per week. How can weekl revenue be maimized? 4. Graph Quadratic Functions in Standard Form 239

4. EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 5, 37, and 57 5 TAKS PRACTICE AND REASONING Es. 39, 40, 43, 53, 58, 60, 62, and 63 5 MULTIPLE REPRESENTATIONS E. 59. VOCABULARY Cop and complete: The graph of a quadratic function is called a(n)?. 2. WRITING Describe how to determine whether a quadratic function has a minimum value or a maimum value. EXAMPLE on p. 236 for Es. 3 2 USING A TABLE Cop and complete the table of values for the function. 3. 5 4 2 4. 523 2 22 2 0 2????? 22 2 0 2????? 5. 5 } 2 2 6. 52 } 3 2 24 22 0 2 4????? 26 23 0 3 6????? MAKING A GRAPH Graph the function. Compare the graph with the graph of 5 2. 7. 5 3 2 8. 5 5 2 9. 522 2 0. 52 2. f() 5 } 3 2 2. g() 52 } 4 2 EXAMPLE 2 on p. 237 for Es. 3 8 3. 5 5 2 4. 5 4 2 5. f() 52 2 2 6. g() 522 2 2 5 7. f() 5 3 } 4 2 2 5 8. g() 52 } 5 2 2 2 ERROR ANALYSIS Describe and correct the error in analzing the graph of 5 4 2 24 2 7. 9. The -coordinate of the verte is: 5 b } 2a 5 24 } 2(4) 5 3 20. The -intercept of the graph is the value of c, which is 7. EXAMPLE 3 on p. 238 for Es. 2 32 MAKING A GRAPH Graph the function. Label the verte and ais of smmetr. 2. 5 2 2 22. 5 3 2 2 6 4 23. 524 2 8 2 24. 522 2 2 6 3 25. g() 52 2 2 2 2 26. f() 526 2 2 4 2 5 27. 5 2 } 3 2 2 3 6 28. 52 3 } 4 2 2 4 2 29. g() 52 3 } 5 2 2 2 30. f() 5 } 2 2 2 3 3. 5 8 } 5 2 2 4 5 32. 52 5 } 3 2 2 2 4 240 Chapter 4 Quadratic Functions and Factoring

EXAMPLE 4 on p. 239 for Es. 33 38 MINIMUMS OR MAXIMUMS Tell whether the function has a minimum value or a maimum value. Then find the minimum or maimum value. 33. 526 2 2 34. 5 9 2 7 35. f() 5 2 2 8 7 36. g() 523 2 8 2 5 37. f() 5 3 } 2 2 6 4 38. 52 } 4 2 2 7 2 39. TAKS REASONING What is the effect on the graph of the function 5 2 2 when it is changed to 5 2 2 3? A The graph widens. B The graph narrows. C The graph opens down. D The verte moves down the -ais. 40. TAKS REASONING Which function has the widest graph? A 5 2 2 B 5 2 C 5 0.5 2 D 52 2 IDENTIFYING COEFFICIENTS In Eercises 4 and 42, identif the values of a, b, and c for the quadratic function. 4. The path of a basketball thrown at an angle of 458 can be modeled b 520.02 2 6. 42. The path of a shot put released at an angle of 358 can be modeled b 520.0 2 0.7 6. 358 43. TAKS REASONING Write three different quadratic functions whose graphs have the line 5 4 as an ais of smmetr but have different -intercepts. MATCHING In Eercises 44 46, match the equation with its graph. 44. 5 0.5 2 2 2 45. 5 0.5 2 3 46. 5 0.5 2 2 2 3 A. (2, 5) (0, 3) B. (0, 3) (2, ) C. 2 (0, 0) (2, 22) MAKING A GRAPH Graph the function. Label the verte and ais of smmetr. 47. f() 5 0. 2 2 48. g() 520.5 2 2 5 49. 5 0.3 2 3 2 50. 5 0.25 2 2.5 3 5. f() 5 4.2 2 6 2 52. g() 5.75 2 2 2.5 53. TAKS REASONING The points (2, 3) and (24, 3) lie on the graph of a quadratic function. Eplain how these points can be used to find an equation of the ais of smmetr. Then write an equation of the ais of smmetr. 54. CHALLENGE For the graph of 5 a 2 b c, show that the -coordinate of the verte is 2} b2 c. 4a 4. Graph Quadratic Functions in Standard Form 24

PROBLEM SOLVING EXAMPLE 5 on p. 239 for Es. 55 58 55. ONLINE MUSIC An online music store sells about 4000 songs each da when it charges $ per song. For each $.05 increase in price, about 80 fewer songs per da are sold. Use the verbal model and quadratic function to find how the store can maimize dail revenue. Revenue (dollars) 5 Price (dollars/song) p Sales (songs) R() 5 ( 0.05) p (4000 2 80) 56. DIGITAL CAMERAS An electronics store sells about 70 of a new model of digital camera per month at a price of $320 each. For each $20 decrease in price, about 5 more cameras per month are sold. Write a function that models the situation. Then tell how the store can maimize monthl revenue from sales of the camera. 57. GOLDEN GATE BRIDGE Each cable joining the two towers on the Golden Gate Bridge can be modeled b the function 5 } 9000 2 2 7 } 5 500 where and are measured in feet. What is the height h above the road of a cable at its lowest point? 58. TAKS REASONING A woodland jumping mouse hops along a parabolic path given b 520. 2 2.3 where is the mouse s horizontal position (in feet) and is the corresponding height (in feet). Can the mouse jump over a fence that is 3 feet high? Eplain. 59. MULTIPLE REPRESENTATIONS A communit theater sells about 50 tickets to a pla each week when it charges $20 per ticket. For each $ decrease in price, about 0 more tickets per week are sold. The theater has fied epenses of $500 per week. a. Writing a Model Write a verbal model and a quadratic function to represent the theater s weekl profit. b. Making a Table Make a table of values for the quadratic function. c. Drawing a Graph Use the table to graph the quadratic function. Then use the graph to find how the theater can maimize weekl profit. 242 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS

60. TAKS REASONING In 97, astronaut Alan Shepard hit a golf ball on the moon. The path of a golf ball hit at an angle of 458 and with a speed of 00 feet per second can be modeled b 52 g } 0,000 2 where is the ball s horizontal position (in feet), is the corresponding height (in feet), and g is the acceleration due to gravit (in feet per second squared). a. Model Use the information in the diagram to write functions for the paths of a golf ball hit on Earth and a golf ball hit on the moon. GRAPHING CALCULATOR In part (b), use the calculator s zero feature to answer the questions. b. Graphing Calculator Graph the functions from part (a) on a graphing calculator. How far does the golf ball travel on Earth? on the moon? c. Interpret Compare the distances traveled b a golf ball on Earth and on the moon. Your answer should include the following: a calculation of the ratio of the distances traveled a discussion of how the distances and values of g are related 6. CHALLENGE Lifeguards at a beach want to rope off a rectangular swimming section. The have P feet of rope with buos. In terms of P, what is the maimum area that the swimming section can have? MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson.2; TAKS Workbook 62. TAKS PRACTICE Liz s high score in a video game is 200 points less than three times her friend s high score. Let represent her friend s high score. Which epression can be used to determine Liz s high score? TAKS Obj. 2 A 200 2 3 B 2 200 } 3 C } 3 2 200 D 3 2 200 REVIEW Lesson.3; TAKS Workbook 63. TAKS PRACTICE The total cost, c, of a school banquet is given b c 5 25n 400, where n is the total number of students attending the banquet. The total cost of the banquet was $9900. How man students attended the banquet? TAKS Obj. 4 F 77 G 340 H 396 J 452 EXTRA PRACTICE for Lesson 4., p. 03 ONLINE QUIZ at classzone.com 243

Graphing Calculator ACTIVITY 4. Find Maimum and Minimum Values Use after Lesson 4. TEKS a.5, a.6 TEXAS classzone.com Kestrokes QUESTION How can ou use a graphing calculator to find the maimum or minimum value of a function? EXAMPLE Find the maimum value of a function Find the maimum value of 522 2 2 0 2 5 and the value of where it occurs. STEP Graph function Graph the given function and select the maimum feature. STEP 2 Choose left bound Move the cursor to the left of the maimum point. Press. CALCULATE :value 2:zero 3:minimum 4:maimum 5:intersect 6:d/d Left Bound? X=-3.4042 Y=5.8646 STEP 3 Choose right bound Move the cursor to the right of the maimum point. Press. STEP 4 Find maimum Put the cursor approimatel on the maimum point. Press. Right Bound? X=-.4893 Y=5.4572 Maimum X=-2.5 Y=7.5 c The maimum value of the function is 5 7.5 and occurs at 522.5. P RACTICE Tell whether the function has a maimum value or a minimum value. Then find the maimum or minimum value and the value of where it occurs.. 5 2 2 6 4 2. f() 5 2 2 3 3 3. 523 2 9 2 4. 5 0.5 2 0.8 2 2 5. h() 5 } 2 2 2 3 2 6. 52 3 } 8 2 6 2 5 244 Chapter 4 Quadratic Functions and Factoring