Characteristics of Quadratic Functions

Transcription

1 . Characteristics of Quadratic Functions Essential Question What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr? Parabolas and Smmetr Work with a partner. a. Complete the table. Then use the values in the table to sketch the graph of the function f() = 1 on graph paper f() 3 5 f() b. Use the results in part (a) to identif the verte of the parabola. c. Find a vertical line on our graph paper so that when ou fold the paper, the left portion of the graph coincides with the right portion of the graph. What is the equation of this line? How does it relate to the verte? d. Show that the verte form f() = 1 ( ) is equivalent to the function given in part (a). ATTENDING TO PRECISION To be proficient in math, ou need to use clear definitions in our reasoning and discussions with others. Parabolas and Smmetr Work with a partner. Repeat Eploration 1 for the function given b f() = = 1 3 ( 3) +. Communicate Your Answer 3. What tpe of smmetr does the graph of f() = a( h) + k have and how can ou describe this smmetr?. Describe the smmetr of each graph. Then use a graphing calculator to verif our answer. a. f() = ( 1) + b. f() = ( + 1) c. f() = ( 3) + 1 d. f() = 1 ( + ) e. f() = + 3 f. f() = 3( 5) + Section. Characteristics of Quadratic Functions 55

2 . Lesson What You Will Learn Core Vocabular ais of smmetr, p. 5 standard form, p. 5 minimum value, p. 58 maimum value, p. 58 intercept form, p. 59 Previous -intercept Eplore properties of parabolas. Find maimum and minimum values of quadratic functions. Graph quadratic functions using -intercepts. Solve real-life problems. Eploring Properties of Parabolas An ais of smmetr is a line that divides a parabola into mirror images and passes through the verte. Because the verte of f() = a( h) + k is (h, k), the ais of smmetr is the vertical line = h. Previousl, ou used transformations to graph quadratic functions in verte form. You can also use the ais of smmetr and the verte to graph quadratic functions written in verte form. (h, k) = h Using Smmetr to Graph Quadratic Functions Graph f() = ( + 3) +. Label the verte and ais of smmetr. SOLUTION Step 1 Identif the constants a =, h = 3, and k =. Step Plot the verte (h, k) = ( 3, ) and draw the ais of smmetr = 3. Step 3 Evaluate the function for two values of. = : f( ) = ( + 3) + = = 1: f( 1) = ( 1 + 3) + = Plot the points (, ), ( 1, ), and their reflections in the ais of smmetr. Step Draw a parabola through the plotted points. ( 3, ) = 3 Quadratic functions can also be written in standard form, f() = a + b + c, where a 0. You can derive standard form b epanding verte form. f() = a( h) + k Verte form f() = a( h + h ) + k Epand ( h). f() = a ah + ah + k f() = a + ( ah) + (ah + k) Distributive Propert Group like terms. 5 Chapter Quadratic Functions f() = a + b + c Let b = ah and let c = ah + k. This allows ou to make the following observations. a = a: So, a has the same meaning in verte form and standard form. b = ah: Solve for h to obtain h = b a. So, the ais of smmetr is = b a. c = ah + k: In verte form f() = a( h) + k, notice that f(0) = ah + k. So, c is the -intercept.

3 Core Concept Properties of the Graph of f() = a + b + c = a + b + c, a > 0 = a + b + c, a < 0 (0, c) = b a b = a (0, c) The parabola opens up when a > 0 and opens down when a < 0. The graph is narrower than the graph of f() = when a > 1 and wider when a < 1. The ais of smmetr is = b a ( and the verte is b a (, f a) b ). The -intercept is c. So, the point (0, c) is on the parabola. Graphing a Quadratic Function in Standard Form COMMON ERROR Be sure to include the negative sign when writing the epression for the -coordinate of the verte. Graph f() = Label the verte and ais of smmetr. SOLUTION Step 1 Identif the coefficients a = 3, b =, and c = 1. Because a > 0, the parabola opens up. Step Find the verte. First calculate the -coordinate. = b a = (3) = 1 Then find the -coordinate of the verte. f(1) = 3(1) (1) + 1 = So, the verte is (1, ). Plot this point. Step 3 Draw the ais of smmetr = 1. Step Identif the -intercept c, which is 1. Plot the point (0, 1) and its reflection in the ais of smmetr, (, 1). Step 5 Evaluate the function for another value of, (1, )) such as = 3. = 1 f(3) = 3(3) (3) + 1 = 10 Plot the point (3, 10) and its reflection in the ais of smmetr, ( 1, 10). Step Draw a parabola through the plotted points. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Label the verte and ais of smmetr. 1. f() = 3( + 1). g() = ( ) h() = + 1. p() = Section. Characteristics of Quadratic Functions 57

4 STUDY TIP When a function f is written in verte form, ou can use h = b a and k = f ( a) b to state the properties shown. Finding Maimum and Minimum Values Because the verte is the highest or lowest point on a parabola, its -coordinate is the maimum value or minimum value of the function. The verte lies on the ais of smmetr, so the function is increasing on one side of the ais of smmetr and decreasing on the other side. Core Concept Minimum and Maimum Values For the quadratic function f() = a + b + c, the -coordinate of the verte is the minimum value of the function when a > 0 and the maimum value when a < 0. a > 0 a < 0 decreasing increasing minimum = b a Minimum value: f ( a) b Domain: All real numbers Range: f ( a) b increasing = b a maimum decreasing Maimum value: f ( a) b Domain: All real numbers Range: f ( a) b Decreasing to the left of = b a Increasing to the right of = b a Increasing to the left of = b a Decreasing to the right of = b a Check Minimum X= Y= Finding a Minimum or a Maimum Value Find the minimum value or maimum value of f() = 1 1. Describe the domain and range of the function, and where the function is increasing and decreasing. SOLUTION Identif the coefficients a = 1, b =, and c = 1. Because a > 0, the parabola opens up and the function has a minimum value. To find the minimum value, calculate the coordinates of the verte. = b a = ( 1 ) = f() = 1 () () 1 = 3 The minimum value is 3. So, the domain is all real numbers and the range is 3. The function is decreasing to the left of = and increasing to the right of =. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. Find the minimum value or maimum value of (a) f() = and (b) h() = Describe the domain and range of each function, and where each function is increasing and decreasing. 58 Chapter Quadratic Functions

5 REMEMBER An -intercept of a graph is the -coordinate of a point where the graph intersects the -ais. It occurs where f() = 0. COMMON ERROR Remember that the -intercepts of the graph of f() = a( p)( q) are p and q, not p and q. Graphing Quadratic Functions Using -Intercepts When the graph of a quadratic function has at least one -intercept, the function can be written in intercept form, f() = a( p)( q), where a 0. Core Concept Properties of the Graph of f() = a( p)( q) Because f(p) = 0 and f(q) = 0, p and = q are the -intercepts of the graph of the function. The ais of smmetr is halfwa between (p, 0) and (q, 0). So, the ais of smmetr is = p + q. The parabola opens up when a > 0 and opens down when a < 0. Graphing a Quadratic Function in Intercept Form Graph f() = ( + 3)( 1). Label the -intercepts, verte, and ais of smmetr. SOLUTION Step 1 Identif the -intercepts. The -intercepts are p = 3 and q = 1, so the parabola passes through the points ( 3, 0) and (1, 0). Step Find the coordinates of the verte. = p + q = = 1 f( 1) = ( 1 + 3)( 1 1) = 8 So, the ais of smmetr is = 1 and the verte is ( 1, 8). Step 3 Draw a parabola through the verte and the points where the -intercepts occur. (p, 0) p + q (q, 0) ( 1, 8) ( 3, 0) = a( p)( q) = 1 (1, 0) Check You can check our answer b generating a table of values for f on a graphing calculator. -intercept -intercept X X=-1 Y The values show smmetr about = 1. So, the verte is ( 1, 8). Monitoring Progress Help in English and Spanish at BigIdeasMath.com Graph the function. Label the -intercepts, verte, and ais of smmetr.. f() = ( + 1)( + 5) 7. g() = 1 ( )( ) Section. Characteristics of Quadratic Functions 59

6 Solving Real-Life Problems Modeling with Mathematics (50, 5) The parabola shows the path of our first golf shot, where is the horizontal distance (in ards) and is the corresponding height (in ards). The path of our second shot can be modeled b the function f() = 0.0( 80). Which shot travels farther before hitting the ground? Which travels higher? SOLUTION (0, 0) (100, 0) 1. Understand the Problem You are given a graph and a function that represent the paths of two golf shots. You are asked to determine which shot travels farther before hitting the ground and which shot travels higher.. Make a Plan Determine how far each shot travels b interpreting the -intercepts. Determine how high each shot travels b finding the maimum value of each function. Then compare the values. 3. Solve the Problem First shot: The graph shows that the -intercepts are 0 and 100. So, the ball travels 100 ards before hitting the ground. 5 d 100 d Because the ais of smmetr is halfwa between (0, 0) and (100, 0), the ais of smmetr is = = 50. So, the verte is (50, 5) and the maimum height is 5 ards. Second shot: B rewriting the function in intercept form as f() = 0.0( 0)( 80), ou can see that p = 0 and q = 80. So, the ball travels 80 ards before hitting the ground. To find the maimum height, find the coordinates of the verte. = p + q = = 0 f(0) = 0.0(0)(0 80) = 3 The maimum height of the second shot is 3 ards. 0 = 5 Because 100 ards > 80 ards, the first shot travels farther. Because 3 ards > 5 ards, the second shot travels higher. 0 0 f 90. Look Back To check that the second shot travels higher, graph the function representing the path of the second shot and the line = 5, which represents the maimum height of the first shot. The graph rises above = 5, so the second shot travels higher. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 8. WHAT IF? The graph of our third shot is a parabola through the origin that reaches a maimum height of 8 ards when = 5. Compare the distance it travels before it hits the ground with the distances of the first two shots. 0 Chapter Quadratic Functions

7 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check 1. WRITING Eplain how to determine whether a quadratic function will have a minimum value or a maimum value.. WHICH ONE DOESN T BELONG? The graph of which function does not belong with the other three? Eplain. f() = 3 + f() = 3 + f() = 3( )( + ) f() = 3( + 1) 7 Monitoring Progress and Modeling with Mathematics In Eercises 3 1, graph the function. Label the verte and ais of smmetr. (See Eample 1.) 3. f() = ( 3). h() = ( + ) 5. g() = ( + 3) + 5. = ( 7) 1 7. = ( ) + 8. g() = ( + 1) 3 9. f() = ( 1) h() = ( + ) + REASONING In Eercises 19 and 0, use the ais of smmetr to plot the reflection of each point and complete the parabola (, 3) 1 (1, ) (0, 1) = 0. = 3 ( 1, 1) (, ) ( 3, 3) 11. = 1 ( + ) = 1 ( 3) f() = 0.( 1) 1. g() = ANALYZING RELATIONSHIPS In Eercises 15 18, use the ais of smmetr to match the equation with its graph. 15. = ( 3) = ( + ) 17. = 1 ( + 1) = ( ) 1 A. C. = = 3 B. D. = 1 = In Eercises 1 30, graph the function. Label the verte and ais of smmetr. (See Eample.) 1. = = = f() = g() = 1. f() = 5 7. g() = f() = = = WRITING Two quadratic functions have graphs with vertices (, ) and (, 3). Eplain wh ou can not use the aes of smmetr to distinguish between the two functions. 3. WRITING A quadratic function is increasing to the left of = and decreasing to the right of =. Will the verte be the highest or lowest point on the graph of the parabola? Eplain. Section. Characteristics of Quadratic Functions 1

8 ERROR ANALYSIS In Eercises 33 and 3, describe and correct the error in analzing the graph of = The -coordinate of the verte is = b a = () = 3. In Eercises 39 8, find the minimum or maimum value of the function. Describe the domain and range of the function, and where the function is increasing and decreasing. (See Eample 3.) 39. = 1 0. = =. g() = The -intercept of the graph is the value of c, which is f() = g() = h() = 1. h() = MODELING WITH MATHEMATICS In Eercises 35 and 3, is the horizontal distance (in feet) and is the vertical distance (in feet). Find and interpret the coordinates of the verte. 35. The path of a basketball thrown at an angle of 5 can be modeled b = The path of a shot put released at an angle of 35 can be modeled b = = f() = PROBLEM SOLVING The path of a diver is modeled b the function f() = , where f() is the height of the diver (in meters) above the water and is the horizontal distance (in meters) from the end of the diving board. a. What is the height of the diving board? b. What is the maimum height of the diver? c. Describe where the diver is ascending and where the diver is descending ANALYZING EQUATIONS The graph of which function has the same ais of smmetr as the graph of = + +? A = + + B = 3 + C = + D = USING STRUCTURE Which function represents the widest parabola? Eplain our reasoning. A = ( + 3) B = 5 C = 0.5( 1) + 1 D = PROBLEM SOLVING The engine torque (in foot-pounds) of one model of car is given b = , where is the speed (in thousands of revolutions per minute) of the engine. a. Find the engine speed that maimizes torque. What is the maimum torque? b. Eplain what happens to the engine torque as the speed of the engine increases. MATHEMATICAL CONNECTIONS In Eercises 51 and 5, write an equation for the area of the figure. Then determine the maimum possible area of the figure w b w b Chapter Quadratic Functions

9 In Eercises 53 0, graph the function. Label the -intercept(s), verte, and ais of smmetr. (See Eample.) 53. = ( + 3)( 3) 5. = ( + 1)( 3) 55. = 3( + )( + ) 5. f() = ( 5)( 1) 57. g() = ( + ) 58. = ( + 7) 59. f() = ( 3) 0. = ( 7) USING TOOLS In Eercises 1, identif the -intercepts of the function and describe where the graph is increasing and decreasing. Use a graphing calculator to verif our answer. 1. f() = 1 ( )( + ). = 3 ( + 1)( 3) 3. g() = ( )( ). h() = 5( + 5)( + 1) 5. MODELING WITH MATHEMATICS A soccer plaer kicks a ball downfield. The height of the ball increases until it reaches a maimum height of 8 ards, 0 ards awa from the plaer. A second kick is modeled b = ( ). Which kick travels farther before hitting the ground? Which kick travels higher? (See Eample 5.). MODELING WITH MATHEMATICS Although a football field appears to be flat, some are actuall shaped like a parabola so that rain runs off to both sides. The cross section of a field can be modeled b = ( 10), where and are measured in feet. What is the width of the field? What is the maimum height of the surface of the field? 8. OPEN-ENDED Write two different quadratic functions in intercept form whose graphs have the ais of smmetr = PROBLEM SOLVING An online music store sells about 000 songs each da when it charges $1 per song. For each $0.05 increase in price, about 80 fewer songs per da are sold. Use the verbal model and quadratic function to determine how much the store should charge per song to maimize dail revenue. Revenue (dollars) = Price (dollars/song) Sales (songs) R() = ( ) (000 80) 70. PROBLEM SOLVING An electronics store sells 70 digital cameras per month at a price of $30 each. For each $0 decrease in price, about 5 more cameras per month are sold. Use the verbal model and quadratic function to determine how much the store should charge per camera to maimize monthl revenue. Revenue (dollars) = Price (dollars/camera) Sales (cameras) R() = (30 0) (70 + 5) 71. DRAWING CONCLUSIONS Compare the graphs of the three quadratic functions. What do ou notice? Rewrite the functions f and g in standard form to justif our answer. f() = ( + 3)( + 1) g() = ( + ) 1 h() = USING STRUCTURE Write the quadratic function f() = + 1 in intercept form. Graph the function. Label the -intercepts, -intercept, verte, and ais of smmetr. surface of football field Not drawn to scale 73. PROBLEM SOLVING A woodland jumping mouse hops along a parabolic path given b = , where is the mouse s horizontal distance traveled (in feet) and is the corresponding height (in feet). Can the mouse jump over a fence that is 3 feet high? Justif our answer. 7. REASONING The points (, 3) and (, ) lie on the graph of a quadratic function. Determine whether ou can use these points to find the ais of smmetr. If not, eplain. If so, write the equation of the ais of smmetr. Not drawn to scale Section. Characteristics of Quadratic Functions 3

10 7. HOW DO YOU SEE IT? Consider the graph of the function f() = a( p)( q). 77. MAKING AN ARGUMENT The point (1, 5) lies on the graph of a quadratic function with ais of smmetr = 1. Your friend sas the verte could be the point (0, 5). Is our friend correct? Eplain. 78. CRITICAL THINKING Find the -intercept in terms of a, p, and q for the quadratic function f() = a( p)( q). a. What does f ( p + q ) represent in the graph? b. If a < 0, how does our answer in part (a) change? Eplain. 79. MODELING WITH MATHEMATICS A kernel of popcorn contains water that epands when the kernel is heated, causing it to pop. The equations below represent the popping volume (in cubic centimeters per gram) of popcorn with moisture content (as a percent of the popcorn s weight). Hot-air popping: = 0.71( 5.5)(.) Hot-oil popping: = 0.5( 5.35)( 1.8) 75. MODELING WITH MATHEMATICS The Gateshead Millennium Bridge spans the River Tne. The arch of the bridge can be modeled b a parabola. The arch reaches a maimum height of 50 meters at a point roughl 3 meters across the river. Graph the curve of the arch. What are the domain and range? What do the represent in this situation? 7. THOUGHT PROVOKING You have 100 feet of fencing to enclose a rectangular garden. Draw three possible designs for the garden. Of these, which has the greatest area? Make a conjecture about the dimensions of the rectangular garden with the greatest possible area. Eplain our reasoning. a. For hot-air popping, what moisture content maimizes popping volume? What is the maimum volume? b. For hot-oil popping, what moisture content maimizes popping volume? What is the maimum volume? c. Use a graphing calculator to graph both functions in the same coordinate plane. What are the domain and range of each function in this situation? Eplain. 80. ABSTRACT REASONING A function is written in intercept form with a > 0. What happens to the verte of the graph as a increases? as a approaches 0? Maintaining Mathematical Proficienc Solve the equation. Check for etraneous solutions. (Skills Review Handbook) = 0 8. = = = + Solve the proportion. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons = 8. 3 = = = 0 Chapter Quadratic Functions