Graphing Quadratic Functions

Transcription

1 Graphing Quadratic Functions. Graphing = a. Focus of a Parabola. Graphing = a + c. Graphing = a + b + c. Comparing Linear, Eponential, and Quadratic Functions What tpe of graph is this? Sorr, no it s the nd degree. ges the graph of = + c. Good, now when I change c = to c = ou jump up. Copright Big Ideas Learning, LLC

2 What You Learned Before Eample Graph =. Step : Make a table of values. Oka, Descartes, this will test the theor of parabolic flight paths. = (, ) = ( ) (, ) = () (, ) = () (, ) = () (, ) Step : Plot the ordered pairs. Step : Draw a line through the points. (, ) (, ) (, ) (, ) Graph the linear equation.. =. = +. = + Eample Evaluate + when =. + = ( ) + ( ) Substitute for. = () + ( ) Evaluate the power. = Multipl. = Subtract. Evaluate the epression when = Copright Big Ideas Learning, LLC

3 . Graphing = a What are the characteristics of the graph of the quadratic function = a? How does the value of a affect the graph of = a? ACTIVITY: Graphing a Quadratic Function Work with a partner. Complete the input-output table. Plot the points in the table. Sketch the graph b connecting the points with a smooth curve. What do ou notice about the graphs? a. = b. = COMMON CORE Graphing Quadratic Functions In this lesson, ou will identif characteristics of quadratic functions. graph quadratic functions. Learning Standard F.BF Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

4 ACTIVITY: Graphing a Quadratic Function Math Practice Look for Patterns What pattern do ou notice when comparing each equation with its graph? Work with a partner. Graph each function. How does the value of a affect the graph of = a? a. = b. = 9 7 c. =. d. = 9 7. IN YOUR OWN WORDS What are the characteristics of the graph of the quadratic function = a? How does the value of a affect the graph of = a? Consider a <, a >, and < a < in our answer. Use what ou learned about the graphs of quadratic functions to complete Eercises 7 on page 7. Copright Big Ideas Learning, LLC Section. Graphing = a

5 . Lesson Lesson Tutorials Ke Vocabular quadratic function, p. parabola, p. verte, p. ais of smmetr, p. A quadratic function is a nonlinear function that can be written in the standard form = a + b + c, where a. The U-shaped graph of a quadratic function is called a parabola. Characteristics of Quadratic Functions The most basic quadratic function is =. The lowest or highest point on a parabola is the verte. decreasing Ais of smmetr increasing Verte The vertical line that divides the parabola into two smmetric parts is the ais of smmetr. The ais of smmetr passes through the verte. EXAMPLE Identifing Characteristics of a Quadratic Function Consider the graph of the quadratic function. Using the graph, ou can identif the verte, ais of smmetr, and the behavior of the graph as shown. You can also determine the following: The domain is all real numbers. The range is all real numbers greater than or equal to. When <, increases as decreases. When >, increases as increases. Function is decreasing. Ais of smmetr: Function is increasing. Verte: (, ) Eercises Identif characteristics of the graph of the quadratic function Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

6 Graphing = a When a > When < a <, the graph of = a opens up and is wider than the graph of =. When a >, the graph of = a opens up and is narrower than the graph of =. a a a Graphing = a When a < When < a <, the graph of = a opens down and is wider than the graph of =. When a <, the graph of = a opens down and is narrower than the graph of =. a a a EXAMPLE Graphing = a When a > Graph =. Compare the graph to the graph of =. Step : Make a table of values. Step : Plot the ordered pairs. Step : Draw a smooth curve through the points. Both graphs open up and have the same verte, (, ), and the same ais of smmetr, =. The graph of = is narrower than the graph of =. 7 Eercises Graph the function. Compare the graph to the graph of =.. =. =. = Copright Big Ideas Learning, LLC Section. Graphing = a

7 EXAMPLE Graphing = a When a < Graph =. Compare the graph to the graph of =. Step : Make a table of values. Choose -values that make the calculations simple. Ź Ź Ź â Ź Step : Plot the ordered pairs. â Step : Draw a smooth curve through the points. Ź The graphs have the same verte, (, ), and the same ais of Ź smmetr, =, but the graph of = opens down. The Ź Ź graph of = is wider than the graph of =. Ź EXAMPLE Real-Life Application The diagram shows the cross section of a satellite dish, where and are measured in meters. Find the width and depth of the dish. Use the domain of the function to find the width of the dish. Use the range to find the depth. (Ź, ) Ź Ź â (, ) (, ) width depth The leftmost point on the graph is (, ) and the rightmost point is (, ). So, the domain is, which represents meters. The lowest point on the graph is (, ) and the highest points on the graph are (, ) and (, ). So, the range is, which represents meter. So, the satellite dish is meters wide and meter deep. Graph the function. Compare the graph to the graph of =. Eercises and. = 7. =.. = 9. The cross section of a spotlight can be modeled b the graph of =., where and are measured in inches and. Find the width and depth of the spotlight. Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions Copright Big Ideas Learning, LLC // :: PM

8 . Eercises Help with Homework. VOCABULARY Describe the verte and ais of smmetr of the graph of = a.. VOCABULARY What is the U-shaped graph of a quadratic function called?. WRITING Without graphing, which graph is wider, = or =? Eplain our reasoning.. WRITING When does the graph of a quadratic function open up? open down? 9+(-)= +(-)= +(-9)= 9+(-)= Use a graphing calculator to graph the function. Compare the graph to the graph of =.. =.. =. 7. =. Identif characteristics of the graph of the quadratic function Graph the function. Compare the graph to the graph of =.. =. =. =. =. =. = 7 7. WATERFALL A fish swims over the waterfall. The distance (in feet) that the fish falls is given b the function = t, where t is the time (in seconds). ft a. Describe the domain and range of the function. b. Graph the function using the domain in part (a). c. Use the graph to determine when the fish lands in the water below. Copright Big Ideas Learning, LLC Section. Graphing = a 7

9 Graph the function. Compare the graph to the graph of =.. = 9. = 7. =. =. =. = 9. ERROR ANALYSIS Describe and correct the error in graphing and comparing = and =... The graphs have the same verte and the same ais of smmetr. The graph of =. is narrower than the graph of =. Describe the possible values of a.. 7 a. a 7. a. REASONING A parabola opens up and passes through (, ) and (, ). How do ou know that (, ) is not the verte? 9. REASONING Describe the domain and range of the function = a when (a) a > and (b) a <. Determine whether the statement is alwas, sometimes, or never true. Eplain our reasoning.. The graph of = a is narrower than the graph of = when a >.. The graph of = a is narrower than the graph of = when a >.. The graph of = a is wider than the graph of = when < a <.. The graph of = a is wider than the graph of = d when a > d.. BRIDGE The arch support of a bridge can be modeled b =., where and are measured in feet. Find the height and width of the arch. Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

10 In Eercises 7, use the graph of the function = a.. When is the function increasing? When is the function decreasing? 7. Find the value of a when the graph passes through (, ). a, a a, a. MODELING The diagram shows the cross section of a swirling glass of water, where and are measured in centimeters. The surface of the cross section of the rotating liquid is a parabola. (,.) (,.) a. About how wide is the mouth of the glass? b. Suppose the rotational speed of the liquid changes. The cross section can now be modeled b =.. Did the rotational speed increase or decrease? Eplain our reasoning. (, ) 9. ASSEMBLY LINE The number of units an assembl line can produce in hour can be modeled b the function =., where is the number of emploees. The assembl line has a capacit of emploees. a. Describe the domain of the function. b. Graph the function using the domain in part (a). c. Is it better for the compan to run one assembl line at full capacit or two assembl lines at half capacit? Eplain our reasoning.. Logic Is the -intercept of the graph of = a alwas? Justif our answer. Solve the proportion using the Cross Products Propert.. =. = n 9. b = (Skills Review Handbook). 9 = 7. MULTIPLE CHOICE What is the completel factored form of? (Section 7.9) A ( ) B ( )( + ) C ( + ) D ( ) Copright Big Ideas Learning, LLC Section. Graphing = a 9

11 . Focus of a Parabola reflectors have parabolic shapes? Wh do satellite dishes and spotlight ACTIVITY: A Propert of Satellite Dishes Work with a partner. Ras are coming straight t down. When the hit the parabola, the reflect off at the same angle at which the entered. Receiver Draw the outgoing part of each ra so that it intersects the -ais. What do ou notice about where the reflected ras intersect the -ais? Where is the receiver for the satellite dish? Eplain. Satellite Dish Ra Ra Ra COMMON CORE Graphing Quadratic Functions In this lesson, ou will find the foci of parabolas. write equations of parabolas with vertices at the origin given the foci. Learning Standard F.IF. Incoming angle Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

12 ACTIVITY: A Propert of Spotlights Work with a partner. Beams of light are coming from the bulb in a spotlight. When the beams hit the parabola, the reflect off at the same angle at which the entered. Draw the outgoing part of each beam. What do the have in common? Eplain. Math Practice Justif Conclusions â What information can ou use to justif our conclusion? Beam Bulb Incoming angle Beam Ź Ź Beam. IN YOUR OWN WORDS Wh do satellite dishes and spotlight reflectors have parabolic shapes?. Design and draw a parabolic satellite dish. Label the dimensions of the dish. Label the receiver. Use what ou learned about parabolas to complete Eercises on page. Copright Big Ideas Learning, LLC MSCC_ALG_PE_.indd Section. Focus of a Parabola // :: PM

13 . Lesson Lesson Tutorials Ke Vocabular focus, p. The Focus of a Parabola The focus of a parabola is a fied point on the interior of a parabola that lies on the ais of smmetr. A parabola wraps around the focus. For functions of the form = a, the focus is (, a). Ais of smmetr Focus Verte Focus Focus a, a a, a EXAMPLE Finding the Focus of a Parabola Graph =. Identif the focus. Step : Make a table of values. Then graph. Step : Identif the focus. The function is of the form = a, so a =. (, ) a = ( ) Substitute for a. =, or Multipl. So, the focus of the function = is (, ). Eercises 7 Graph the function. Identif the focus.. =. =. = Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

14 EXAMPLE Writing an Equation of a Parabola Write an equation of the parabola with focus (, ) and verte at the origin. For = a, the focus is (, a). Use the given focus, (, ), to write an equation to find a. a = Equate the -coordinates. EXAMPLE = a Multipl each side b a. = a Divide each side b. An equation of the parabola is =. Real-Life Application A birdwatcher uses a parabolic microphone to collect and record bird sounds. The cross section of the microphone can be modeled b =, where and are measured in inches. The focus is located at the end of the receiver arm. What is the length of the receiver arm? The arm length is the distance from the focus to the verte. Identif the focus. For the function =, a =. a = ( = ) Substitute Multipl. = Divide. for a. The focus is (, ). The verte is (, ). The distance from (, ) to (, ) is units. So, the length of the receiver arm is inches. Receiver arm Eercises 9. Write an equation of the parabola with focus (, ) and verte at the origin.. WHAT IF? In Eample, the cross section of the microphone can be modeled b =. What is the length of the receiver arm? Copright Big Ideas Learning, LLC Section. Focus of a Parabola

15 . Eercises Help with Homework. VOCABULARY What is the relationship between the focus and the ais of smmetr of a parabola?. WRITING When the focus of a parabola lies below the verte, does the parabola open up or down? Is a > or a <? Eplain.. OPEN-ENDED Write an equation of a parabola whose focus is below the -ais. 9+(-)= +(-)= +(-9)= 9+(-)= Determine whether the shape is parabolic.... Graph the function. Identif the focus. 7. =. = 9. =. =. =. =.7. ERROR ANALYSIS Describe and correct the error in identifing the focus. Write an equation of the parabola with a verte at the origin and the given focus.. (, ). (, ). (, ) ) 7. (,. (, ) 9. (,.). COMET A comet travels along a parabolic path around the Sun. The Sun is the focus of the path. When the comet is at the verte of the path, it is,, kilometers from the Sun. Write an equation that represents the path of the comet. Assume the focus is on the positive -ais and the verte is (, )., ) ) Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

16 Match the equation with its graph.. =. =. =. A. B. C. Focus Focus Focus Determine whether the statement is sometimes, alwas, or never true for the function = a. Eplain our reasoning.. The verte and focus of a parabola can occur at the same point.. If a parabola opens down, then the focus lies below the -ais.. The -coordinate of the focus of a parabola is greater than the -coordinate of the verte. 7. REASONING Describe how the graph of = a changes as the distance between the verte and focus increases.. WHISPER DISH Whisper dishes are parabolic sound reflectors that transmit and receive sound waves from opposite ends of a room. For the best sound reception, ou place our ear at the focus of the dish. Write an equation for the cross section of a dish when our ear is feet from the verte. 9. SOLAR COOKING You make a solar cooker using a parabolic reflective surface. You suspend a sausage with wire through the focus of each end piece of the cooker. How far from the bottom should ou place the wire?. For what values of a will the distance between the focus and the verte of the graph of = a be less than the distance between the focus and the verte of the graph of = (a)? in. in. Evaluate the epression when = and n =. (Skills Review Handbook) n. + n. MULTIPLE CHOICE Which number is equivalent to /? (Section.) A B C D Copright Big Ideas Learning, LLC Section. Focus of a Parabola

17 . Graphing = a + c of = a + c? How does the value of c affect the graph ACTIVITY: Graphing = a + c Work with a partner. Sketch the graphs of both functions in the same coordinate plane. How does the value of c affect the graph of = a + c? a. = and = + b. = and = c. = + and = + 9 d. = and = COMMON CORE Graphing Quadratic Functions In this lesson, ou will graph quadratic functions of the form = a + c and compare to the graph of =. Learning Standard F.BF Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

18 Math Practice Communicate Precisel What have ou included in our answer to make sure our eplanation is precise? ACTIVITY: Finding -Intercepts of Graphs Work with a partner. Graph each function. Find the -intercepts of the graph. Eplain how ou found the -intercepts. a. = b. = 7 7 c. = + d. = 7 7. IN YOUR OWN WORDS How does the value of c affect the graph of = a + c? Use a graphing calculator to verif our conclusions. Use what ou learned about the graphs of quadratic functions to complete Eercises 7 9 on page. Copright Big Ideas Learning, LLC Section. Graphing = a + c 7

19 Lesson. Lesson Tutorials Ke Vocabular zero, p. 9 Graphing = + c When c >, the graph of = + c is a vertical translation c units up of the graph of =. EXAMPLE c When c <, the graph of = + c is a vertical translation c units down of the graph of =. câ c Graphing = + c Graph =. Compare the graph to the graph of =. Step : Make a table of values. Step : Plot the ordered pairs. â Ź Ź Ź Step : Draw a smooth curve through the points. Both graphs open up and have the same ais of smmetr, =. The graph of = is a translation units down of the graph of =. â Ź Ź EXAMPLE Graphing = a + c Graph = +. Compare the graph to the graph of =. Step : Make a table of values. Step : Plot the ordered pairs. 7 7 Step : Draw a smooth curve through the points. â â à Both graphs open up and have the same ais of smmetr, =. The graph of = + is narrower than the graph of =. The verte of the graph of = + is a translation unit up of the verte of the graph of =. Graph the function. Compare the graph to the graph of =. Eercises 7 Chapter MSCC_ALG_PE_.indd. = + Graphing Quadratic Functions. =. = + Copright Big Ideas Learning, LLC // :: PM

20 EXAMPLE Translating the Graph of = + c Which of the following is true when ou translate the graph of = to the graph of = +? A The graph shifts units up. B The graph shifts 7 units up. C The graph shifts 7 units down. D The graph shifts units down. Both graphs open up and have the same ais of smmetr, =. The verte of = is (, ). The verte of = + is (, ). To move the verte from (, ) to (, ), ou must translate the graph 7 units up. The correct answer is B. A zero of a function f () is an -value for which f () =. A zero is located at the -intercept of the graph of the function. EXAMPLE Real-Life Application The function f (t) = t + s gives the approimate height (in feet) of a falling object t seconds after it is dropped from an initial height s (in feet). An egg is dropped from a height of feet. When does the egg hit the ground? ft The initial height is feet. So, the function f (t) = t + gives the height of the egg after t seconds. It hits the ground when f (t) =. Step : Make a table of values and sketch the graph. Common Error The graph in Eample shows the height of the object over time, not the path of the object. f(t) â Źt à t f (t ) Step : Find the zero of the function. When t =, f (t) =. So, the zero is. t The egg hits the ground seconds after it is dropped. Eercises,. The graph of = + is shifted to =. Describe the translation.. REASONING Eplain wh onl nonnegative values of t are used in Eample.. WHAT IF? In Eample, the egg is dropped from a height of feet. When does the egg hit the ground? Copright Big Ideas Learning, LLC MSCC_ALG_PE_.indd 9 Section. Graphing = a + c 9 // :: PM

21 Eercises. Help with Homework. VOCABULARY Describe the verte and ais of smmetr of the graph of = a + c. How is the value of c related to the verte of the graph?. NUMBER SENSE Without graphing, which graph has the greater -intercept, = + or =? Eplain our reasoning.. WRITING How does the graph of = a + c compare to the graph of = a? )= 9+(- )= +(- 9)= +(- = ) 9+(- Match each function with its graph.. = + A.. = + B.. = C. 7 Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Graph the function. Compare the graph to the graph of =. 7. = +. =. = 9. = 9. = +. = + Use a graphing calculator to graph the function. Compare the graph to the graph of =.. = +. =. = Describe how to translate the graph of = + to the graph of the given function.. = + 7. = 9. ERROR ANALYSIS Describe and correct the error in comparing the graphs.. REASONING The domain of = a + c is all real numbers. Describe the range when (a) a > and (b ) a <.. WATER BALLOON A water balloon is dropped from a height of feet. The function h = + gives the height h of the balloon after seconds. When does it hit the ground? Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions. =. â â à Ź Ź The graph of = + is a translation units down of the graph of =. Copright Big Ideas Learning, LLC // :: PM

22 . APPLE The function = + gives the height (in feet) of an apple after falling seconds. Find and interpret the - and -intercepts. Find the zeros of the function.. =. =. = + 9 Sketch a quadratic function with the given characteristics.. The parabola opens up and the verte is (, ). 7. The parabola opens down, the verte is (, ), and one of the -intercepts is.. The function is increasing when < and the -intercepts of the parabola are and. 9. The graph is below the -ais and the highest point on the parabola is (, ).. REASONING Describe two algebraic methods ou can use to find the zeros of the function f (t ) = t +.. REASONING Can the focus and the verte of a parabola lie on opposite sides of the -ais? Eplain our reasoning.. PROBLEM SOLVING The paths of water from three different garden waterfalls are given below. Each function gives the height h (in feet) and the horizontal distance d (in feet) of the water. Waterfall : h =.d +. Waterfall : h =.d +.9 Waterfall : h =.d +. a. Which waterfall drops water from the highest point? b. Which waterfall follows the narrowest path? c. Which waterfall sends water the farthest?. Logic Let f () be a quadratic function of the form f () = a + c. a. How does the graph of f () + k compare to the graph of f () when k <? when k >? b. Let k be a real number not equal to or. How does the graph of k f () compare to the graph of f ()? Factor the polnomial. (Section 7.7 and Section 7.) MULTIPLE CHOICE What is the product of ( ) and ( )? (Section 7.) A B + C D Copright Big Ideas Learning, LLC Section. Graphing = a + c

23 Stud Help Graphic Organizer You can use a summar triangle to eplain a topic. Here is an eample of a summar triangle for graphing a quadratic function of the form = a when a >. Graphing = a when a > When < a <, the graph of = a is wider than the graph of =. Eample: a = < a < When a >, the graph of = a is narrower than the graph of =. Eample: a = a > Make summar triangles to help ou stud these topics.. graphing = a when a <. identifing the focus of a parabola. graphing = a + c After ou complete this chapter, make summar triangles for the following topics.. graphing = a + b + c. graphing = a( h) + k What do ou call a cheese summar triangle that isn t ours? Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

24 .. Quiz Progress Check Identif characteristics of the graph of the quadratic function. (Section.).... Graph the function. Compare the graph to the graph of =. (Section.). =. = 7. = Graph the function. Identif the focus. (Section.). = 9. =. = Write an equation of the parabola with a verte at the origin and the given focus. (Section.). (, ). (, ). (, Graph the function. Compare the graph to the graph of =. (Section.). = +. =. = + 7. PINEAPPLE The distance (in feet) that a pineapple falls is given b the function = t, where t is the time (in seconds). Use a graph to determine how man seconds it takes for the pineapple to fall feet. (Section.). SMARTPHONE A new smartphone application is available for download. The number of downloads can be modeled b the function =. +, where is the number of hours since the new application was released. How man hours does it take for the number of downloads to reach? (Section.) ) Copright Big Ideas Learning, LLC Sections.. Quiz

25 . Graphing = a + b + c = a + b + c? How can ou find the verte of the graph of ACTIVITY: Comparing Two Graphs Work with a partner. Sketch the graphs of = and = +. What do ou notice about the -value of the verte of each graph? = = COMMON CORE Graphing Quadratic Functions In this lesson, ou will find the aes of smmetr and the vertices of graphs. graph quadratic functions of the form = a + b + c. find maimum and minimum values. Learning Standards F.BF. F.IF. F.IF.7a ACTIVITY: Comparing -Intercepts with the Verte Work with a partner. Use the graph in Activit to find the -intercepts of the graph of =. Verif our answer b solving =. Compare the location of the verte to the location of the -intercepts. Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

26 ACTIVITY: Finding Intercepts Math Practice Use Prior Results How can ou use results from the previous activities to complete the table? Work with a partner. Solve = a + b b factoring. What are the -intercepts of the graph of = a + b? Cop and complete the table to verif our answer. = a + b b a ACTIVITY: Deductive Reasoning Work with a partner. Complete the following logical argument. The -intercepts of the graph of = a + b are and b a. The verte of the graph of = a + b occurs when =. The vertices of the graphs of = a + b and = a + b + c have the same -value. The verte of = a + b + c occurs when =.. IN YOUR OWN WORDS How can ou find the verte of the graph of = a + b + c?. Without graphing, find the verte of the graph of = +. Check our result b graphing. Use what ou learned about the vertices of the graphs of quadratic functions to complete Eercises on page 9. Copright Big Ideas Learning, LLC Section. Graphing = a + b + c

27 . Lesson Lesson Tutorials Ke Vocabular maimum value, p. 7 minimum value, p. 7 Properties of the Graph of = a + b + c The graph opens up when a > and the graph opens down when a <. The -intercept is c. The -coordinate of the âź b a (, c) The ais of smmetr b a is =. EXAMPLE â a à b à c, where a verte is. b a verte Finding the Ais of Smmetr and the Verte of a Graph Find (a) the ais of smmetr and (b) the verte of the graph of = +. a. Find the ais of smmetr when a = and b =. b a Write the equation for the ais of smmetr. = () Substitute for a and for b. = Simplif. = The ais of smmetr is =. b. The ais of smmetr is =, so the -coordinate of the verte is. Use the function to find the -coordinate of the verte. = + Write the function. = ( ) + ( ) Substitute for. = 9 Simplif. The verte is (, 9). Eercises Find (a) the ais of smmetr and (b) the verte of the graph of the function.. = Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions. = + +. = + 7 Copright Big Ideas Learning, LLC // :: PM

28 EXAMPLE Graphing = a + b + c Graph = +. Describe the domain and range. Step : Find and graph the ais of smmetr. = b a = ( ) () = Step : Find and plot the verte. The ais of smmetr is =, so the -coordinate of the verte is. Use the function to find the -coordinate of the verte. = + Write the function. 7 = () () + Substitute for. = Simplif. So, the verte is (, ). Step : Use the -intercept to find two more points on the graph. The -intercept is. So, (, ) lies on the graph. Because the ais of smmetr is =, the point (, ) also lies on the graph. Step : Draw a smooth curve through the points. The domain is all real numbers. The range is. Eercises Graph the function. Describe the domain and range.. = + +. = + 7. = Maimum and Minimum Values The -coordinate of the verte of the graph of = a + b + c is the maimum value of the function when a < or the minimum value of the function when a >. = a + b + c, a < = a + b + c, a > maimum minimum Copright Big Ideas Learning, LLC Section. Graphing = a + b + c 7

29 EXAMPLE Finding Maimum and Minimum Values Tell whether the function f () = 9 has a minimum value or a maimum value. Then find the value. For f () = 9, a = and <. So, the parabola opens down and the function has a maimum value. To find the maimum value, find the -coordinate of the verte. b a ( ) ( ) b a = = = The -coordinate of the verte is. f ( ) = ( ) ( ) 9 Substitute for. = 7 Simplif. The maimum value is 7. Tell whether the function has a minimum value or a maimum value. Then find the value. Eercises 7. g() = + EXAMPLE. h() = + + Real-Life Application The function f (t ) = t + t + gives the height (in feet) of a water balloon t seconds after it is launched. Use a graphing calculator to find the maimum height of the water balloon. Step : Enter the function f (t) = t + t + into our calculator and graph it. Because time cannot be negative, use onl nonnegative values of t. f(t) â Źt à t à Step : Use the maimum feature to find the maimum value of the function. Stud Tip The minimum feature of a graphing calculator can be used for parabolas that open up. The maimum height of the water balloon is feet. 9. When does the water balloon reach its maimum height? Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions Copright Big Ideas Learning, LLC // :: PM

30 . Eercises Help with Homework. VOCABULARY Eplain how ou can tell whether a quadratic function has a maimum value or a minimum value without graphing the function.. DIFFERENT WORDS, SAME QUESTION Consider the quadratic function = + +. Which is different? Find both answers. What is the maimum value of the function? What is the -coordinate of the verte of the graph? What is the greatest number in the range of the function? What is the ais of smmetr of the graph of the function? 9+(-)= +(-)= +(-9)= 9+(-)= Find the verte, the ais of smmetr, and the -intercept of the graph Find (a) the ais of smmetr and (b) the verte of the graph of the function.. = 7. = +. = 9. = +. = +. = +. ERROR ANALYSIS Describe and correct the error in finding the ais of smmetr of the graph of = +. = b a = () = The ais of smmetr is =. Graph the function. Describe the domain and range.. = + +. = + +. = 9. = = + 9. = Copright Big Ideas Learning, LLC Section. Graphing = a + b + c 9

31 9. FIREWORK The function shown represents the height h (in feet) of a firework t seconds after it is launched. The firework eplodes at its highest point. h â Źt à t a. When does the firework eplode? b. Describe the domain and range of h.. REASONING Given the quadratic equation = a + b + c, find the ais of smmetr when b =. Tell whether the function has a minimum value or a maimum value. Then find the value.. = +. = = +. = + +. = +. = PRECISION The verte of a graph of a quadratic function is (, ). One point on the graph is (, ). Find another point on the graph. Justif our answer.. SUSPENSION BRIDGE The cables between the two towers of a suspension bridge can be modeled b = +, where and are measured in feet. The cables are at road level midwa between the towers. How high is the road above the water? Ź à â 9. STEEPLECHASE The function h(t ) = t + t gives the height h (in feet) of a horse t seconds after it jumps during a steeplechase. a. When does the horse reach its maimum height? b. Can the horse clear a fence that is. feet tall? If so, b how much? c. How long is the horse in the air? Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions Copright Big Ideas Learning, LLC // :: PM

32 Use the minimum or maimum feature of a graphing calculator to approimate the verte of the graph of the function.. =. +.. =. +. = π +. CHOOSE TOOLS The graph of a quadratic function passes through (, ), (, ), and (, ). Does the graph open up or down? Eplain our reasoning.. REASONING For a quadratic function f, what does f ( a) b represent? Eplain our reasoning.. CALCULATORS An office suppl store sells about graphing calculators per month for $ each. For each $ decrease in price, the store epects to sell more calculators. a. Write a quadratic function that represents the revenue from calculator sales. (Note: revenue = units sold unit price) b. How much should the store charge per calculator to maimize monthl revenue?. AIR CANNON At a basketball game, an air cannon is used to launch T-shirts into the crowd. The function = + gives the path of a T-shirt. The function = gives the height of the bleachers. In both functions, represents height (in feet) and represents horizontal distance (in feet). At what height does the T-shirt land in the bleachers? w Pla area Dog shelter building w 7. DOG SHELTER The owners of a dog shelter want to enclose e a rectangular pla area on the side of their building. The have k feet of fencing. What is the maimum area of the outside enclosure in terms of k? (Hint: Find the -coordinate of the verte of the graph of the area function.) Graph the function. (Section. and Section.). + = 9. = (.) t. r(t ) = (.) t. MULTIPLE CHOICE What is the value of () when =? (Section.) A B C D Copright Big Ideas Learning, LLC Section. Graphing = a + b + c

33 Etension. Graphing = a( h) + k Lesson Tutorials Ke Vocabular verte form, p. The verte form of a quadratic function is = a( h) + k, where a. The verte of the parabola is (h, k ). Graphing = ( h) When h >, the graph of = ( h) is a horizontal translation h units to the right of the graph of =. h When h <, the graph of = ( h) is a horizontal translation h units to the left of the graph of =. h h EXAMPLE Graphing = ( h) COMMON CORE Graphing Quadratic Functions In this etension, ou will graph quadratic functions of the form = a( h) + k and compare to the graph of =. Learning Standards F.BF. F.IF. F.IF.7a Graph = ( ). Compare the graph to the graph of =. Step : Make a table of values. Step : Plot the ordered pairs. Step : Draw a smooth curve through the points. The graph of = ( ) is a translation units to the right of the graph of =. ( ) Graph the function. Compare the graph to the graph of =. Use a graphing calculator to check our answer.. = ( + ). = ( ). = ( ). = ( + ). = (.). = ( + 7. REASONING Compare the graphs of = and = without graphing the functions. How can factoring help ou compare the parabolas? Eplain.. STRUCTURE Write the function in Eample in the form = a + b + c. Describe advantages and disadvantages of writing the function in each form. ) Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

34 EXAMPLE Graphing = ( h) + k Graph = ( + ). Compare the graph to the graph of =. 7 Step : Make a table of values. 7 Step : Plot the ordered pairs. ( ) Step : Draw a smooth curve through the points. The graph of = ( + ) is a translation units to the left and unit down of the graph of =. EXAMPLE Graphing = a ( h) + k Graph = ( + ) +. Compare the graph to the graph of =. Stud Tip Notice what the values in verte form represent: a: opens up or down, and is wider or narrower h: horizontal translation k: vertical translation (h, k): verte Step : Make a table of values. Step : Plot the ordered pairs. Step : Draw a smooth curve through the points. ( ) The graph of = ( + ) + opens down and is narrower than the graph of =. The verte of the graph of = ( + ) + is a translation units to the left and units up of the verte of the graph of =. Graph the function. Compare the graph to the graph of =. Use a graphing calculator to check our answer. 9. = ( ) +. = ( + ) 7. = ( ). = ( ) +. = ( ). = ( + ) Describe how the graph of g() compares to the graph of f ().. g() = f () 7. g() = f ( + ) 7. g() = f (). g() = f () 9. REASONING The graph of = is translated units right and units down. Write an equation for the function in verte form and in standard form.. REASONING Does k represent the -intercept of the graph of = a( h) + k? Eplain. Copright Big Ideas Learning, LLC Etension. Graphing = a( h) + k

35 . Comparing Linear, Eponential, and Quadratic Functions linear, eponential, and quadratic functions? How can ou compare the growth rates of ACTIVITY: Comparing Speeds Work with a partner. Three cars start traveling at the same time. The distance traveled in t minutes is miles. Complete each table and sketch all three graphs in the same coordinate plane. Compare the speeds of the three cars. Which car has a constant speed? Which car is accelerating the most? Eplain our reasoning. t = t t = t t = t COMMON CORE Linear, Quadratic, and Eponential Functions In this lesson, ou will identif linear, quadratic, and eponential functions using graphs or tables. Learning Standards F.IF. F.IF. F.IF.7a F.LE. Distance (miles) Time (minutes). t Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

36 ACTIVITY: Comparing Speeds Math Practice Analze Relationships What is the relationship between the speeds of the cars? Work with a partner. Analze the speeds of the three cars over the given time periods. The distance traveled in t minutes is miles. Which car eventuall overtakes the others? a. t = t t = t t = t b. t = t t = t t = t IN YOUR OWN WORDS How can ou compare the growth rates of linear, eponential, and quadratic functions? Which tpe of growth eventuall leaves the other two in the dust? Eplain our reasoning. Use what ou learned about comparing functions to complete Eercises on page 9. Copright Big Ideas Learning, LLC Section. Comparing Linear, Eponential, and Quadratic Functions

37 . Lesson Lesson Tutorials Linear Function Eponential Function Quadratic Function Line = m + b EXAMPLE Curve = ab Parabola = a + b + c Identifing Functions Using Graphs Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function. a. (, ), (, ), (, ) ( ),, (, ) b. (, ), (, ), (, 7), (, ), (, ) ( 7 Ź Ź Ź Ź Ź Ź Ź Ź Ź Quadratic function ) (, ),, Eercises 9 c. (, ), (, ), (, ), Ź Ź Ź Linear function Eponential function Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function.. (, ), (, ), (, ), (, ), (, ) ( )( ). (, ), (, ), (, ),,,,. (, ), (, ), (, ), (, 7), (, ) Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions Copright Big Ideas Learning, LLC // :9: PM

38 Differences and Ratios of Functions Eponential Function: = () Linear Function: = The -values have a common difference of. The -values have a common ratio of. Quadratic Function: = Stud Tip EXAMPLE + + For a linear function, the first differences are constant First differences + + Second differences For quadratic functions, the second differences are constant. Identifing Functions Using Differences or Ratios Tell whether the table of values represents a linear, an eponential, or a quadratic function. Stud Tip For a quadratic function, the -values will increase, then decrease, or the -values will decrease, then increase. Eercises b The -values have a common difference of. So, the table represents a linear function.. Tell whether the table of values represents a linear, an eponential, or a quadratic function. Copright Big Ideas Learning, LLCSection MSCC_ALG_PE_.indd a The second differences are constant. So, the table represents a quadratic function. 9 7 Comparing Linear, Eponential, and Quadratic Functions 7 // :9: PM

39 EXAMPLE 9 Identifing and Writing a Function Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. Step : Graph the data. The function appears to be eponential or quadratic. Step : Check the -values. If there is no common difference or ratio, check the second differences Ź Stud Tip + 9 Second differences are constant The function is quadratic. Step : Use the form = a. To check our function in Eample, substitute the other points from the table to see if the satisf the function. = a() Use the point (, ). Substitute for and for. Solve for a. =a So, an equation for the quadratic function is =.. Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. Eercises 9 Linear Function Eponential Function = ab, a, b, and b > = m + b m a b : Eponential deca m Chapter MSCC_ALG_PE_.indd Graphing Quadratic Functions Quadratic Function = a + b + c, a a a b : Eponential growth a Copright Big Ideas Learning, LLC // :9:7 PM

40 . Eercises Help with Homework. VOCABULARY How can ou decide whether to use a linear, a quadratic, or an eponential function to model a data set?. WHICH ONE DOESN T BELONG? Which graph does not belong with the other three? Eplain our reasoning. f() g() m() n() 9+(-)= +(-)= +(-9)= 9+(-)= Find the values of when f () is greater than g().. f() =. f() =. f() = g() = g() = g() = Match the function tpe with its graph.. Linear function 7. Eponential function. Quadratic function A. 7 B. C. Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function. 9. (, ), (, ), (, ), (, ), (, ). (, ), (, ) (,, ), (, ), (, ). (, ), (, ), (, 9), (, 9), (, ). (, ), (, ), (, ), (, ), (, ). SUBWAY A student takes a subwa to a public librar. The table shows the distance d (in miles) the student travels in t minutes. Tell whether Time, t Distance, d the data can be modeled b a linear, an eponential, or a quadratic function. Copright Big Ideas Learning, LLC Section. Comparing Linear, Eponential, and Quadratic Functions 9

41 Tell whether the table of values represents a linear, an eponential, or a quadratic function REASONING Can the -values of a data set have both a common difference and a common ratio? Eplain our reasoning. Tell whether the data values represent a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. 9. (, ), (, ), (, ), (, ), (, )..... Ź Ź Ź Ź. (, ), (, ), (, ), (, ), (,.) Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź Ź. ERROR ANALYSIS Describe and correct the error in writing an equation for the function represented b the ordered pairs. (, ), (, ), (, ), (, ), (, ) Chapter MSCC_ALG_PE_.indd The ordered pairs represent a quadratic function. + + Graphing Quadratic Functions = a = a() First differences Second differences =a So, an equation is =. Copright Big Ideas Learning, LLC // :9: PM

42 . HIGH SCHOOL FOOTBALL The table shows the number of people attending the first five football games at a high school. a. Plot the points. b. Does a linear, an eponential, or a quadratic function represent this situation? Eplain. Game, g Number of People, p CRITICAL THINKING Is the graph of a set of points enough to determine whether the points represent a linear, an eponential, or a quadratic function? Justif our answer.. RECORDING STUDIO The table shows the amount of mone (in dollars) that a musician pas for using a recording studio. a. Plot the points. Then determine the tpe of function that best represents this situation. b. Write a function that models the data. Number of Hours, h Amount, m (dollars) c. How much does it cost to use the studio for hours? 9. TOURNAMENT At the beginning of a basketball tournament, there are teams. After each round, one-half of the remaining teams are eliminated. a. Make a table showing the number of teams remaining after each round. b. Determine the tpe of function that best represents this situation. c. Write a function that models the data. d. After which round do ou know the team that won the tournament?. Repeated Reasoning Write a function that has constant second differences of. Find the -intercept(s) of the graph. (Section. and Section.).... MULTIPLE CHOICE What is the factored form of? (Section 7.9) A ( + ) B ( + )( ) C ( ) D ( + )( ) Copright Big Ideas Learning, LLC Section. Comparing Linear, Eponential, and Quadratic Functions

43 Etension. Comparing Graphs of Functions Lesson Tutorials You have alread learned that the average rate of change (or slope) between an two points on a line is the change in divided b the change in. You can find the average rate of change between two points of a nonlinear function using the same method. ACTIVITY Rates of Change of a Quadratic Function In Eample on page, the function f (t ) = t + t + gives the height (in feet) of a water balloon t seconds after it is launched. COMMON CORE Linear, Quadratic, and Eponential Functions In this etension, ou will compare graphs of linear, quadratic, and eponential functions. solve real-life problems. Learning Standards F.IF. F.IF. F.IF.7a F.LE. a. Cop and complete the table for f (t ). t..... f (t ) b. Graph the ordered pairs from part (a). Then draw a smooth curve through the points. c. For what values is the function increasing? decreasing? d. Cop and complete the tables to find the average rate of change for each interval. 9 7 t Time Interval to. sec. to sec to. sec. to sec to. sec Average Rate of Change (ft/sec) Time Interval. to sec to. sec. to sec to. sec. to sec Average Rate of Change (ft/sec). Compared to the average rate of change of a linear function, what do ou notice about the average rate of change in part (d) of Activit?. Is the average rate of change increasing or decreasing from to. seconds? How can ou use the graph to justif our answer?. What do ou notice about the average rate of change when the function is increasing and when the function is decreasing?. In Eample on page 9, the function f (t) = t + gives the height of an egg t seconds after it is dropped. (a) Make a table of values. Use the domain t with intervals of. second. (b) Graph the ordered pairs and draw a smooth curve through the points. (c) Describe where the function is increasing and decreasing. (d) Find the average rate of change for each interval in the table. What do ou notice? Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

44 ACTIVITY Rates of Change of Different Functions The graphs show the numbers of videos on three video-sharing websites hours after the websites are launched. Linear Quadratic Eponential Number of videos Website (, ) (, ) (, ) (, ) (, ) (, ) (, ) Time (hours) Number of videos Website (, ) (, ) (, ) (, ) (, ) (, ) (, ) Time (hours) Number of videos Website (, 9) (, ) (, ) (, ) (, ) (, ) (, ) Time (hours) a. Do the three websites ever have the same number of videos? b. Cop and complete the table for each function. Time Interval to h to h to h to h to h to h Average Rate of Change (videos/hour) c. What do ou notice about the average rate of change of the linear function? d. What do ou notice about the average rate of change of the quadratic function? e. What do ou notice about the average rate of change of the eponential function? f. Which average rate of change increases more quickl, the quadratic function or the eponential function?. REASONING How does a quantit that is increasing eponentiall compare to a quantit that is increasing linearl or quadraticall?. REASONING Eplain wh the average rate of change of a linear function is constant and the average rate of change of a quadratic or eponential function is not constant. Copright Big Ideas Learning, LLC Etension. Comparing Graphs of Functions

45 .. Quiz Progress Check Find (a) the ais of smmetr and (b) the verte of the graph of the function. (Section.). =. = + + Graph the function. Describe the domain and range. (Section.). = + 7. = +. = + Tell whether the function has a minimum value or a maimum value. Then find the value. (Section.). = + 7. = + +. = + + Graph the function. Compare the graph to the graph of =. (Section.) 9. = ( ). = ( + ) Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function. (Section.). (, ), (, ), (, ), (, ), (, ). (, 7.), (,.), (, ), (, 7), (,.) Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. (Section.) FOOTBALL The function h(t ) = t + t + gives the height (in feet) of a football t seconds after it is thrown. Describe the domain and range. Find the maimum height of the football. (Section.) 7. DOWNLOADING MUSIC The table shows the amounts of mone (in dollars) that ou pa to download songs from a website. (Section.) a. Plot the points. Tell whether the points represent a linear, an eponential, or a quadratic function. b. Write a function that models the data. Number of Songs, s Amount, a (dollars) c. How much does it cost to download songs?..7.. Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

46 Chapter Review Review Ke Vocabular quadratic function, p. parabola, p. verte, p. ais of smmetr, p. focus, p. zero, p. 9 Review Eamples and Eercises. Graphing = a (pp. 9) Vocabular Help maimum value, p. 7 minimum value, p. 7 verte form, p. Graph =. Compare the graph to the graph of =. Step : Make a table of values. Step : Plot the ordered pairs. Step : Draw a smooth curve through the points. The graphs have the same verte, (, ), and the same ais of smmetr, =, but the graph of = opens down. The graph of = is narrower than the graph of =. Graph the function. Compare the graph to the graph of =.. = 7. =. =. Focus of a Parabola (pp. ) Write an equation of the parabola with focus (, ) and verte at the origin. For = a, the focus is (, a). Use the given focus, (, ), to write an equation to find a. a = = a Solve for a. Equate the -coordinates. An equation of the parabola is =. Copright Big Ideas Learning, LLC Chapter Review

47 . Graph =. Identif the focus.. Write an equation of the parabola with focus (, ) and verte at the origin.. Graphing = a + c (pp. ) Graph = +. Compare the graph to the graph of =. Step : Make a table of values. Step : Plot the ordered pairs. Step : Draw a smooth curve through the points. Both graphs open up and have the same ais of smmetr, =. The graph of = + is narrower than the graph of =. The verte of the graph of = + is a translation units up of the verte of the graph of =. Graph the function. Compare the graph to the graph of =.. = + 7. =. =. Graphing = a + b + c (pp. ) Graph = +. Describe the domain and range. Step : Find and graph the ais of smmetr: = b a = () =. Step : Find and plot the verte. The -coordinate of the verte is. The -coordinate is: = ( ) + ( ) =. So, the verte is (, ). Step : Use the -intercept to find two more points on the graph. The -intercept is. So, (, ) lies on the graph. Because the ais of smmetr is =, the point (, ) also lies on the graph. 7 Step : Draw a smooth curve through the points. The domain is all real numbers. The range is. Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

48 Graph the function. Describe the domain and range. 9. = + 7. = +. = +. The function f (t ) = t + 7t + gives the height (in feet) of a pumpkin t seconds after it is launched from a catapult. Use a graphing calculator to find the maimum height of the pumpkin. When does the pumpkin reach its maimum height? Graph the function. Compare the graph to the graph of =. Use a graphing calculator to check our answer.. = ( + ).. = ( + ). = ( ) + Comparing Linear, Eponential, and Quadratic Functions (pp. ) Tell whether the data values represent a linear, an eponential, or a quadratic function. a. (, ), (, ), (, ), (, ), (, ) b Ź Ź Ź Ź Ź 7 The points represent a quadratic function The -values have a common difference of 7. So, the table represents a linear function.. Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a. 7. The function f (t ) = t + 7t + gives the height (in feet) of a baseball t seconds after it is thrown. Sketch a graph of the function. Find the average rate of change from to second and from to seconds. Is the average rate of change increasing or decreasing? How does the graph justif our answer? Copright Big Ideas Learning, LLC MSCC_ALG_PE ec.indd 7 Chapter Review 7 // :: PM

49 Chapter Test Graph the function. Compare the graph to the graph of =. Test Practice. =. = +. =. = ( ). = ( + ). = ( ) Graph the function. Identif the focus. 7. =. = 9. =. Graph the function. Describe the domain and range.. = +. = +. = +. Describe how the graph of g() = f ( + ) compares to the graph of f (). Tell whether the table of values represents a linear, an eponential, or a quadratic function. Then write an equation for the function using the form = m + b, = ab, or = a.... EARTH S ORBIT The table shows the distance d (in miles) that the Earth moves in its orbit around the Sun after t seconds. Tell whether the data can be modeled b a linear, an eponential, or a quadratic function. Time, t Distance, d RADIO TELESCOPE An observator uses a radio telescope to collect data from another gala. The cross section of the telescope s dish can be modeled b =, where and are measured in meters. The telescope s receiver is located at the focus of the parabola. What is the distance from the verte of the parabola to the receiver?. REASONING Consider the function f () = +. Is the average rate of change increasing or decreasing from = to =? Eplain. Chapter Graphing Quadratic Functions Copright Big Ideas Learning, LLC

50 Standards Assessment. What is the equation of the parabola shown in the graph? (F.IF.) Test-Taking Strateg Answer Eas Questions First A. = B. = Scan the test and answer the eas questions first. Because the graph opens up, ou know the answer must be A. C. = D. =. Which epression is equivalent to ( b )? (N.RN.) F. b H. b 9 G. b 9 I. b. What is the ais of smmetr of the graph of =? (F.IF.) A. = C. = B. = D. =. What is the minimum value of the function = + +? (F.IF.7a). What are the solutions of a 9 =? (A.REI.b) F. a = 7 and a = 7 H. a = and a = G. a = 7 and a = 7 I. a = 7 and a = 7 Copright Big Ideas Learning, LLC Standards Assessment 9