Properties of the Graph of a Quadratic Function. has a vertex with an x-coordinate of 2 b } 2a

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1 0.2 Graph 5 a 2 b c Before You graphed simple quadratic functions. Now You will graph general quadratic functions. Wh? So ou can investigate a cable s height, as in Eample 4. Ke Vocabular minimum value maimum value You can use the properties below to graph an quadratic function. You will justif the formula for the ais of smmetr in Eercise 38 on page 639. KEY CONCEPT For Your Notebook Properties of the Graph of a Quadratic Function The graph of 5 a 2 b c is a parabola that: opens up if a. 0 and opens down if a, 0. is narrower than the graph of 5 2 if a. and wider if a,. has an ais of smmetr of 52 b } 2a. has a verte with an -coordinate of 2 b } 2a. has a -intercept of c. So, the point (0, c) is on the parabola. 5 a 2 b c, a > 0 (0, c) 52 b 2a E XAMPLE Find the ais of smmetr and the verte Consider the function a. Find the ais of smmetr of the graph of the function. b. Find the verte of the graph of the function. IDENTIFY THE VERTEX Because the verte lies on the ais of smmetr, 5 3, the -coordinate of the verte is 3. a. For the function , a 522 and b b } 2a } 2(22) 5 3 Substitute 22 for a and 2 for b. Then simplif. b. The -coordinate of the verte is 2 b } 2a, or 3. To find the -coordinate, substitute 3 for in the function and find. 5 2(3) 2 2(3) Substitute 3 for. Then simplif. c The verte is (3, ). 0.2 Graph 5 a 2 b c 635

2 E XAMPLE 2 Graph 5 a 2 b c Graph AVOID ERRORS Be sure to include the negative sign before the fraction when calculating the ais of smmetr. REVIEW REFLECTIONS For help with reflections, see p STEP Determine whether the parabola opens up or down. Because a. 0, the parabola opens up. STEP 2 Find and draw the ais of smmetr: 52} b 2a 52} 26 2(3) 5. STEP 3 Find and plot the verte. The -coordinate of the verte is 2} b 2a, or. To find the -coordinate, substitute for in the function and simplif. 5 3() 2 2 6() 2 52 So, the verte is (, 2). STEP 4 Plot two points. Choose two -values less than the -coordinate of the verte. Then find the corresponding -values STEP 5 Reflect the points plotted in Step 4 in the ais of smmetr. STEP 6 Draw a parabola through the plotted points. (2, ) 3 (0, 2) (2, 2) (3, ) 5 ais of smmetr 3 verte (, 2) at classzone.com GUIDED PRACTICE for Eamples and 2. Find the ais of smmetr and the verte of the graph of the function Graph the function Label the verte and ais of smmetr. KEY CONCEPT For Your Notebook Minimum and Maimum Values For 5 a 2 b c, the -coordinate of the verte is the minimum value of the function if a. 0 or the maimum value of the function if a, 0. 5 a 2 b c, a > 0 5 a 2 b c, a < 0 maimum minimum 636 Chapter 0 Quadratic Equations and Functions

3 E XAMPLE 3 Find the minimum or maimum value Tell whether the function f() has a minimum value or a maimum value. Then find the minimum or maimum value. Because a 523 and 23, 0, the parabola opens down and the function has a maimum value. To find the maimum value, find the verte. 52 b } 2a } 2(23) 522 The -coordinate is 2 b } 2a. f(22) 523(22) 2 2 2(22) c The maimum value of the function is f(22) Substitute 22 for. Then simplif. E XAMPLE 4 Find the minimum value of a function SUSPENSION BRIDGES The suspension cables between the two towers of the Mackinac Bridge in Michigan form a parabola that can be modeled b the graph of where and are measured in feet. What is the height of the cable above the water at its lowest point? The lowest point of the cable is at the verte of the parabola. Find the -coordinate of the verte. Use a and b b } 2a } 2( ) 90 Use a calculator. Substitute 90 for in the equation to find the -coordinate of the verte (90) (90) c The cable is about 96 feet above the water at its lowest point. GUIDED PRACTICE for Eamples 3 and 4 3. Tell whether the function f() has a minimum value or a maimum value. Then find the minimum or maimum value. 4. SUSPENSION BRIDGES The cables between the two towers of the Takoma Narrows Bridge form a parabola that can be modeled b the graph of the equation where and are measured in feet. What is the height of the cable above the water at its lowest point? Round our answer to the nearest foot. 0.2 Graph 5 a 2 b c 637

4 0.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 9 and 4 5 STANDARDIZED TEST PRACTICE Es. 2, 2, 27, 37, 42, and 44. VOCABULARY Eplain how ou can tell whether a quadratic function has a maimum value or minimum value without graphing the function. 2. WRITING Describe the steps ou would take to graph a quadratic function in standard form. EXAMPLE on p. 635 for Es. 3 4 FINDING AXIS OF SYMMETRY AND VERTEX Find the ais of smmetr and the verte of the graph of the function } } } MULTIPLE CHOICE What is the verte of the graph of the function ? A (23, 294) B (23, 24) C (3, 23) D (3, 4) ERROR ANALYSIS Describe and correct the error in finding the ais of smmetr of the graph of the given function } b } 2a 5 6 } 2(2) 5 4 The ais of smmetr is b } 2a } 2 3 } The ais of smmetr is 526. EXAMPLE 2 on p. 636 for Es GRAPHING QUADRATIC FUNCTIONS Graph the function. Label the verte and ais of smmetr } 4 } } } } MULTIPLE CHOICE Which function has the graph shown? A B 52} (0, 3) (2, 5) C 5 } D Chapter 0 Quadratic Equations and Functions

5 EXAMPLE 3 on p. 637 for Es MAXIMUM AND MINIMUM VALUES Tell whether the function has a minimum value or a maimum value. Then find the minimum or maimum value. 28. f() f() f() f() f() f() f() 5 } f() 52 3 } f() 5 } WRITING Compare the graph of with the graph of REASONING Follow the steps below to justif the equation for the ais of smmetr for the graph of 5 a 2 b c. Because the graph of 5 a 2 b c is a vertical translation of the graph of 5 a 2 b, the two graphs have the same ais of smmetr. Use the function 5 a 2 b in place of 5 a 2 b c. a. Find the -intercepts of the graph of 5 a 2 b. (You can do this b finding the zeros of the function 5 a 2 b using factoring.) b. Because a parabola is smmetric about its ais of smmetr, the ais of smmetr passes through a point halfwa between the -intercepts of the parabola. Find the -coordinate of this point. What is an equation of the vertical line through this point? 39. CHALLENGE Write a function of the form 5 a 2 b whose graph contains the points (, 6) and (3, 6). PROBLEM SOLVING GRAPHING CALCULATOR You ma wish to use a graphing calculator to complete the following Problem Solving eercises. EXAMPLE 4 on p. 637 for Es SPIDERS Fishing spiders can propel themselves across water and leap verticall from the surface of the water. During a vertical jump, the height of the bod of the spider can be modeled b the function where is the duration (in seconds) of the jump and is the height (in millimeters) of the spider above the surface of the water. After how man seconds does the spider s bod reach its maimum height? What is the maimum height? 4. ARCHITECTURE The parabolic arches that support the roof of the Dallas Convention Center can be modeled b the graph of the equation where and are measured in feet. What is the height h at the highest point of the arch as shown in the diagram? 20 h Graph 5 a 2 b c 639

6 42. EXTENDED RESPONSE Students are selling packages of flower bulbs to raise mone for a class trip. Last ear, when the students charged $5 per package, the sold 50 packages. The students want to increase the cost per package. The estimate that the will lose 0 sales for each $ increase in the cost per package. The sales revenue R (in dollars) generated b selling the packages is given b the function R 5 (5 n)(50 2 0n) where n is the number of $ increases. a. Write the function in standard form. b. Find the maimum value of the function. c. At what price should the packages be sold to generate the most sales revenue? Eplain our reasoning. 43. AIRCRAFT An aircraft hangar is a large building where planes are stored. The opening of one airport hangar is a parabolic arch that can be modeled b the graph of the equation where and are measured in feet. Graph the function. Use the graph to determine how wide the hangar is at its base SHORT RESPONSE The casts of some Broadwa shows go on tour, performing their shows in cities across the United States. For the period , the number of tickets sold S (in millions) for Broadwa road tours can be modeled b the function S t 2 0.4t 2 where t is the number of ears since 990. Was the greatest number of tickets for Broadwa road tours sold in 995? Eplain. 45. CHALLENGE During an archer competition, an archer shoots an arrow from.5 meters off of the ground. The arrow follows the parabolic path shown and hits the ground in front of the target 90 meters awa. Use the -intercept and the points on the graph to write an equation for the graph that models the path of the arrow. 2 (0,.5) 0 verte (8,.6) (90, 0) MIXED REVIEW Graph the equation. (pp. 25, 225, 244) } Simplif (4 2 2) 2 9 (p. 96) 50. 2( 2 a) 2 5a (p. 96) (22mn) 4 (p. 489) p (7w 7 ) 2 (p. 489) } 24 (p. 03) 6u 3 } v p uv2 } 36 (p. 495) PREVIEW Prepare for Lesson 0.3 in Es Find the zeros of the polnomial function. 55. f() (p. 583) 56. f() (p. 583) 57. f() (p. 593) 58. f() (p. 593) 640 EXTRA PRACTICE for Lesson 0.2, p. 947 ONLINE QUIZ at classzone.com

7 Etension Use after Lesson 0.2 Ke Vocabular intercept form Graph Quadratic Functions in Intercept Form GOAL Graph quadratic functions in intercept form. In Lesson 0.2 ou graphed quadratic functions written in standard form. Quadratic functions can also be written in intercept form, 5 a( 2 p)( 2 q) where a Þ 0. In this form, the -intercepts of the graph can easil be determined. KEY CONCEPT For Your Notebook Graph of Intercept Form 5 a( 2 p)( 2 q) Characteristics of the graph of 5 a( 2 p)( 2 q): The -intercepts are p and q. The ais of smmetr is halfwa between (p, 0) and (q, 0). So, the ais of smmetr is 5 p } q. 2 The parabola opens up if a. 0 and opens down if a, 0. 5 p q 2 (p, 0) (q, 0) E XAMPLE Graph a quadratic function in intercept form FIND ZEROS OF A FUNCTION Notice that the -intercepts of the graph are also the zeros of the function: 0 52( )( 2 5) 5 0 or or 5 5 Graph 52( )( 2 5). STEP Identif and plot the -intercepts. Because p 52 and q 5 5, the -intercepts occur at the points (2, 0) and (5, 0). STEP 2 Find and draw the ais of smmetr. 5 p } q 5 2 } STEP 3 Find and plot the verte. The -coordinate of the verte is 2. To find the -coordinate of the verte, substitute 2 for and simplif. 52(2 )(2 2 5) 5 9 So, the verte is (2, 9). STEP 4 Draw a parabola through the verte and the points where the -intercepts occur. 5 (2, 9) (2, 0) (5, 0) Etension: Graph Quadratic Functions in Intercept Form 64

8 E XAMPLE 2 Graph a quadratic function Graph STEP Rewrite the quadratic function in intercept form Write original function. 5 2( 2 2 4) Factor out common factor. 5 2( 2)( 2 2) Difference of two squares pattern STEP 2 Identif and plot the -intercepts. Because p 522 and q 5 2, the -intercepts occur at the points (22, 0) and (2, 0). STEP 3 Find and draw the ais of smmetr. 5 p } q 5 22 } STEP 4 Find and plot the verte. The -coordinate of the verte is 0. The -coordinate of the verte is: 5 2(0) So, the verte is (0, 28). STEP 5 Draw a parabola through the verte and the points where the -intercepts occur. (22, 0) (2, 0) 2 (0, 28) at classzone.com PRACTICE EXAMPLE on p. 64 for Es. 9 Graph the quadratic function. Label the verte, ais of smmetr, and -intercepts.. 5 ( 2)( 2 3) 2. 5 ( 5)( 2) 3. 5 ( 9) ( 2 5)( ) ( 7)( 2) ( 2 6)( 2 3) } 2 ( 4)( 2 2) 8. 5 ( 2 7)(2 2 3) ( 0)( 2 3) EXAMPLE 2 on p. 642 for Es Follow the steps below to write an equation of the parabola shown. a. Find the -intercepts. b. Use the values of p and q and the coordinates of the verte to find the value of a in the equation 5 a( 2 p)( 2 q). c. Write a quadratic equation in intercept form. 642 Chapter 0 Quadratic Equations and Functions