Quadratic Functions. 2.1 Shape and Structure. 2.2 Function Sense. 2.3 Up and Down. 2.4 Side to Side. 2.5 What s the Point?

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1 Quadratic Functions The Millennium Bridge in London is a bridge solel for pedestrians. It was opened in June of hence its name but was closed for ears for repairs after it got the nickname Wobbl Bridge..1 Shape and Structure Forms of Quadratic Functions Function Sense Translating Functions Up and Down Vertical Dilations of Quadratic Functions Side to Side Horizontal Dilations of Quadratic Functions What s the Point? Deriving Quadratic Functions Now It s Getting Comple... But It s Reall Not Difficult! Comple Number Operations You Can t Spell Fundamental Theorem of Algebra without F-U-N! Quadratics and Comple Numbers

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3 Shape and Structure Forms of Quadratic Functions.1 Learning Goals In this lesson, ou will: Match a quadratic function with its corresponding graph. Identif ke characteristics of quadratic functions based on the form of the function. Analze the different forms of quadratic functions. Use ke characteristics of specific forms of quadratic functions to write equations. Write quadratic functions to represent problem situations. Ke Terms standard form of a quadratic function factored form of a quadratic function verte form of a quadratic function concavit of a parabola Have ou ever seen a tightrope walker? If ou ve ever seen this, ou know that it is quite amazing to witness a person able to walk on a thin piece of rope. However, since safet is alwas a concern, there is usuall a net just in case of a fall. That brings us to a oung French daredevil named Phillippe Petit. Back in 197 with the help of some friends, he spent all night secretl placing a 5 pound cable between the World Trade Center Towers in New York Cit. At dawn, to the shock and amazement of onlookers, the fatigued -ear old Petit stepped out onto the wire. Ignoring the frantic calls of the police, he walked, jumped, laughed, and even performed a dance routine on the wire for nearl an hour without a safet net! Mr. Petit was of course arrested upon climbing back to the safet of the ledge. When asked wh he performed such an unwise, dangerous act, Phillippe said: When I see three oranges, I juggle; when I see two towers, I walk. You can see the events unfold in the Academ Award winning documentar Man on Wire b James Marsh. Have ou ever challenged ourself to do something difficult just to see if ou could do it? 9

4 Problem 1 It s All in the Form 1. Cut out each quadratic function and graph on the net page two pages. a. Tape each quadratic function to its corresponding graph. Please do not use graphing calculators for this activit. What information can ou tell from looking at the function and what can ou tell b looking at each graph? b. Eplain the method(s) ou used to match the functions with their graphs. 9 Chapter Quadratic Functions

5 a. f() 5 ( 1 1)( 1 5) d. f() 5 ( 1) g. f() 5 ( 1 ) b. f() π 1. e. f() 5 ( 1)( 5) h. f() c. f() 5.5( 3)( 3) f. f() i. f() 5 ( 1 ).1 Forms of Quadratic Functions 91

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7 A. B. C. 1 tape function here tape function here tape function here D. E. F tape function here tape function here tape function here G. H. I tape function here tape function here tape function here.1 Forms of Quadratic Functions 93

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9 Recall that quadratic functions can be written in different forms. standard form: f() 5 a 1 b 1 c, where a does not equal. factored form: f() 5 a( r 1 )( r ), where a does not equal. verte form: f() 5 a( h) 1 k, where a does not equal.. Sort our graphs with matching equations into 3 piles based on the function form. Keep these piles; ou will use them again at the end of this Problem. The graphs of quadratic functions can be described using ke characteristics: -intercept(s), -intercept, verte, ais of smmetr, and concave up or down. Concavit of a parabola describes whether a parabola opens up or opens down. A parabola is concave down if it opens downward; a parabola is concave up if it opens upward. 3. The form of a quadratic function highlights different ke characteristics. State the characteristics ou can determine from each. standard form factored form verte form.1 Forms of Quadratic Functions 95

10 . Christine and Kate were asked to determine the verte of two different quadratic functions each written in different forms. Analze their calculations. Christine f() The quadratic function is in standard form. So I know the ais of smmetr is 5 b a ẋ 5 1 () 5 3. Now that I know the ais of smmetr, I can substitute that value into the function to determine the -coordinate of the verte. f(3) 5 (3) 1 1(3) (9) Therefore, the verte is (3, ). Kate g() 5 1_ ( 1 3)( 7) The form of the function tells me the -intercepts are 3 and 7. I also know the -coordinate of the verte will be directl in the middle of the -intercepts. So, all I have to do is calculate the average Now that I know the -coordinate of the verte, I can substitute that value into the function to determine the -coordinate. g() 5 1_ ( 1 3)( 7) 5 1_ (5)(5) Therefore, the verte is (, 1.5). a. How are these methods similar? How are the different? b. What must Kate do to use Christine s method? c. What must Christine do to use Kate s method? 9 Chapter Quadratic Functions

11 5. Analze each table on the following three pages. Paste each function and its corresponding graph from Question in the Graphs and Their Functions section of the appropriate table. Then, eplain how ou can determine each ke characteristic based on the form of the given function. Standard Form f() 5 a 1 b 1 c, where a fi Graphs and Their Functions Methods to Identif and Determine Ke Characteristics Ais of Smmetr -intercept(s) Concavit Verte -intercept.1 Forms of Quadratic Functions 97

12 Factored Form f() 5 a( r 1 )( r ), where a fi Graphs and Their Functions Methods to Identif and Determine Ke Characteristics Ais of Smmetr -intercept(s) Concavit Verte -intercept 9 Chapter Quadratic Functions

13 Verte Form f() 5 a( h) 1 k, where a fi Graphs and Their Functions Methods to Identif and Determine Ke Characteristics Ais of Smmetr -intercept(s) Concavit Verte -intercept.1 Forms of Quadratic Functions 99

14 Problem What Do You Know? 1. Analze each graph. Then, circle the function(s) which could model the graph. Describe the reasoning ou used to either eliminate or choose each function. a. f 1 () 5 ( 1 1)( 1 ) f () f 3 () 5 ( 1 1)( 1 ) f () 5.9 f 5 () 5 ( 1)( ) f () 5 ( ) 1 3 Think about the information given b each function and the relative position of the graph. f 7 () 5 3( 1 )( 3) f () 5 ( 1 ) Chapter Quadratic Functions

15 b. f 1 () 5 ( 75) 9 f () 5 ( )( 1 ) f 3 () 5 1 f () 5 3( 1)( 5) f 5 () 5 ( 75) 9 f () 5 1 f 7 () 5 ( 1 ) f () 5 ( 1 1)( 1 3).1 Forms of Quadratic Functions 11

16 c. f 1 () 5 3( 1 1)( 5) f () 5 ( 1 ) 5 f 3 () 5 1 1,1 f () 5 3( 1 1)( 1 5) f 5 () 5 ( ) 1 5 f () Chapter Quadratic Functions

17 . Consider the two functions shown from Question 1. Identif the form of the function given, and then write the function in the other two forms, if possible. If it is not possible, eplain wh. a. From part (a): f 1 () 5 ( 1 1)( 1 ) b. From part (c): f 5 () 5 ( ) Forms of Quadratic Functions 13

18 Problem 3 Unique... One and Onl? 1. George and Pat were each asked to write a quadratic equation with a verte of (, ). Analze each student s work. Describe the similarities and differences in their equations and determine who is correct. George 5 a( h) 1 k 5 a( ) 1 5 1_ ( ) 1 Pat 5 a( h) 1 k 5 a( ) 1 5 ( ) 1. Consider the 3 forms of quadratic functions and state the number of unknown values in each. Form Number of Unknown Values f() 5 a( h) 1 k f() 5 a( r 1 )( r ) f() 5 a 1 b 1 c a. If a function is written in verte form and ou know the verte, what is still unknown? b. If a function is written in factored form and ou know the roots, what is still unknown? c. If a function is written in an form and ou know one point, what is still unknown? State the unknown values for each form of a quadratic function. 1 Chapter Quadratic Functions

19 d. If ou onl know the verte, what more do ou need to write a unique function? Eplain our reasoning. e. If ou onl know the roots, what more do ou need to write a unique function? Eplain our reasoning. You can write a unique quadratic function given a verte and a point on the parabola. Write the quadratic function given the verte (5, ) and the point (, 9). Substitute the given values into the verte form of the function. Then simplif. Finall, substitute the a-value into the function. f() 5 a( h) 1 k 9 5 a( 5) a(1) a a 7 5 a f() 5 7( 5) 1 You can write a unique quadratic function given the roots and a point on the parabola. Write a quadratic function given the roots (, ) and (, ), and the point (1, ). Substitute the given values into the factored form of the function. f() 5 a( r 1 )( r ) 5 a(1 ())(1 ) Then simplif. Finall, substitute the a-value into the function. 5 a(1 1 )(1 ) 5 a(3)(3) 5 9a 3 5 a f() 5 ( 1 )( ) 3.1 Forms of Quadratic Functions 15

20 3. Eplain wh knowing the verte and a point creates a unique quadratic function.. If ou are given the roots, how man unique quadratic functions can ou write? Eplain our reasoning. 5. Use the given information to determine the most efficient form ou could use to write the function. Write standard form, factored form, verte form, or none in the space provided. a. minimum point (, 75) -intercept (, 15) b. points (, ), (, ), and (, ) c. points (1, 75), (5, 75), and (15, 95) d. points (3, 3), (, 3), and (5, 3) e. -intercepts: (7.9, ) and (7.9, ) point (, ) f. roots: (3, ) and (1, ) point (1, ) g. Ma hits a baseball off a tee that is 3 feet high. The ball reaches a maimum height of feet when it is 15 feet from the tee. h. A grasshopper was standing on the 35 ard line of a football field. He jumped, and landed on the 3 ard line. At the 3 ard line he was inches in the air. 1 Chapter Quadratic Functions

21 Problem Just Another Da at the Circus Write a quadratic function to represent each situation using the given information. Be sure to define our variables. 1. The Amazing Larr is a human cannonball. He would like to reach a maimum height of 3 feet during his net launch. Based on Amazing Larr s previous launches, his assistant DaJuan has estimated that this will occur when he is feet from the cannon. When Amazing Larr is shot from the cannon, he is 1 feet above the ground. Write a function to represent Amazing Larr s height in terms of his distance.. Craz Cornelius is a fire jumper. He is attempting to run and jump through a ring of fire. He runs for 1 feet. Then, he begins his jump just feet from the fire and lands on the other side 3 feet from the fire ring. When Cornelius was 1 foot from the fire ring at the beginning of his jump, he was 3.5 feet in the air. Write a function to represent Craz Cornelius height in terms of his distance. Round to the nearest hundredth. 3. Harsh Knarsh is attempting to jump across an alligator filled swamp. She takes off from a ramp 3 feet high with a speed of 95 feet per second. Write a function to represent Harsh Knarsh s height in terms of time. Remember, the general equation to represent height over time is h(t) 5 1t 1 v t 1 h where v is the initial velocit in feet per second and h is the initial height in feet. Be prepared to share our solutions and methods..1 Forms of Quadratic Functions 17

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23 Function Sense Translating Functions. Learning Goals In this lesson, ou will: Analze the basic form of a quadratic function. Identif the reference points of the basic form of a quadratic function. Understand the structure of the basic quadratic function. Graph quadratic functions through transformations. Identif the effect on a graph b replacing f() b f( C) 1 D. Identif transformations given equations of quadratic functions. Write quadratic functions given a graph. Ke Terms reference points transformation rigid motion argument of a function translation Have ou ever taken a road trip? For most American children, some road trips were the bane of eistence especiall with annoing siblings. Of course, on the drive back, kids might get ecited b a point of reference such as a road sign indicating the number of miles remaining before getting home, or seeing a landscape that is comfortabl familiar. Adults commonl use reference points on road trips as well. Most U.S. national map books give estimated hours of travel and distances between ke cities across the countr. These help motorists and truckers determine if the should continue on, or get off the highwa for a bit of shut ee. What tpes of reference points have ou used? How did ou use those reference points? 19

24 Problem 1 It All Comes Down to the Basics So far in this course, all the different forms of quadratic functions ou have studied have been based on the function f() 5. This function is the basic form of a quadratic function. From the form of this function, ou know the verte is (,). The pattern of this function is that for ever input value,, the output value, f(), is squared. 1. Let s consider the structure of f() 5 and its corresponding graph. a. Complete the table. Then plot and label the points on the coordinate plane. f() 5 1 b. Draw a dashed line to represent the ais of smmetr. Then plot and label all smmetric points. Finall, draw a smooth curve to represent f() Chapter Quadratic Functions

25 Problem Up, Down, Left, Right You just analzed the basic form of a quadratic function, f() 5. For a quadratic function, if ou know the verte and an two points to the right of that verte, ou can use the ais of smmetr to identif the other half of the parabola. A set of ke points that help identif the basic form of an function are called reference points. The reference points of the basic quadratic function are defined in the table shown. Reference Points of the Basic Quadratic Function P (, ) Q (1, 1) R (, ) Now that ou understand the structure of the basic quadratic function, let s eplore how to appl transformations to graph new functions. Recall that a transformation is the mapping, or movement, of all the points of a figure in a plane according to a common operation. Translations, reflections, rotations, and dilations are eamples of transformations. In previous courses, ou studied the effects that transformations had on graphs of functions and various figures on the coordinate plane. A rigid motion is a transformation that preserves size and shape. So, for quadratics I just need to remember the relationship between the verte and two points. To plot Q, I go to the right 1 and up 1 from the verte, and to plot point R, I go to the right and up from the verte. 1. Identif which transformations are rigid motions that preserve size and shape. Previousl, ou worked with the verte form of quadratic functions, f() 5 a( h) 1 k, which represent transformations of f() 5.. Given a quadratic function written in verte form, f() 5 a( h) 1 k, identif the effect the h- and k-values have on the graph of the basic quadratic function. Which transformations are represented b the h- and k-values?. Translating Functions 111

26 Verte form is a specific form of a quadratic function. In this lesson, ou will use the basic quadratic function to eplore various function transformations. Eventuall, ou will learn how to generalize about transformations performed across man different function tpes, but first let s establish a form that will be representative of an function. Transformations performed on an function f() to form a new function g() can be described b the transformational function form: g() 5 Af (B( C)) 1 D where A, B, C, and D represent different constants. Let s make some connections and compare the verte form of a quadratic function, f() 5 a( h) 1 k, to the transformational function form g() 5 Af(B( C)) 1 D, where B How are the h- and k-values of the verte form of the quadratic function represented in the transformational function form? The goal is to understand the effects of transformations using a general function form, and then being able to appl that knowledge to an function famil. Cool! I can learn something once and appl it over and over again to an function tpe. 11 Chapter Quadratic Functions

27 Let s consider the constants C and D, where A and B both equal 1 and the effects each value has on the graph of the basic quadratic function. Let A and B both equal 1. Given f() 5 Graph g() 5 f( ) 1 In the function g(), C 5 and D 5. The C-value and the D-value will tell ou how to translate the function. Notice, the value of is on the inside of the function, or the argument of the function. The argument of a function is the variable, term, or epression on which the function operates. The value is on the outside of the function. Recall values on the inside of a function affect the -values of the function, and values on the outside affect -values of the function. So, the C-value will tell ou how man units to translate the function left or right, and the D-value will tell ou how man units to translate the function up or down. Remember, translations preserve the same size and shape of the function. Given f() 5 Graph g() 5 f( ) 1 You can use reference points for f() and our knowledge about transformations to graph the function g(). From g(), ou know that C 5 and D 5 which tells ou the entire graph will translate units to the right and units up. f() R (, ) P9 ( +, + ) P (, ) Q (1, 1) f() P Q R P9 (, ) R9 ( +, + ) Q9 (1 +, 1 + ) f() g() R9 (, ) Q9 (5, 3) P9 (, ) The function f() is shown with its reference points. To begin to graph g(), plot the new verte, (C, D). This point establishes the new set of aes. Net, think about the pattern of the basic quadratic function. To plot point Q, move right 1 unit, and up 1 unit from the verte. To plot point R, move right units, and up units from the verte. Finall, use smmetr to complete the graph. You can think of the verte of the transformed function as the origin of a new set of aes.. Translating Functions 113

28 . Analze the worked eample. a. Complete the table of values to verif the graph. Reference Points of Basic Quadratic Function Appl the Transformations Corresponding Points on g() P (, ) ( 1, 1 ) P9 (, ) Q (1, 1) (1 1, 1 1 ) Q9 (5, 3) R (, ) ( 1, 1 ) R9 (, ) b. Wh is it important to establish a new set of aes? c. Use C and D to write the equations that correspond to the new set of aes. d. Name the two points smmetric to Q and R. e. The function g() 5 f( ) 1 is written in terms of f(). Rewrite g() in terms of b substituting ( ) for into f(), and adding onto f(). 11 Chapter Quadratic Functions

29 5. Given f() 5, graph h() 5 f( ). a. Identif the C- and D-values. b. Complete the table. Based on the form of h(), what do ou think the graph will look like? Reference Points on f() (, ) (1, 1) (, ) Corresponding Points on h() c. Graph h() in the same 3 steps as the worked eample. Provide the rationale ou used below each graph.. Translating Functions 115

30 . Given f() 5, graph the function m() 5 f( 1 ) 3. Reference Points on f() (, ) Corresponding Points on m() What does the form of m() tell ou? (1, 1) (, ) 7. Analze the graphs of p() and m(). p() m() a. If f() 5, write m() in terms of b. If f() 5, write p() in terms of f() 5. f() 5. c. Write p() in terms of m(). d. Write m() in terms of p(). 11 Chapter Quadratic Functions

31 A translation is a tpe of transformation that shifts an entire figure or graph the same distance and direction. Compared with the graph of 5 f(), the graph of 5 f( C) 1 D: shifts left C units if C,. shifts right C units if C.. shifts down D units if D,. shifts up D units if D... Given 5 f(), write the coordinate notation represented in 5 f( C) 1 D. (, ) Be prepared to share our solutions and methods.. Translating Functions 117

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33 Up and Down Vertical Dilations of Quadratic Functions.3 Learning Goals In this lesson, ou will: Graph quadratic functions through vertical dilations. Identif the effect on a graph b replacing f() b Af(). Write quadratic functions given a graph. Ke Terms vertical dilation vertical stretching vertical compression reflection line of reflection Have ou ever heard that people shrink as the get older? Shrinking can occur from a number of reasons like osteoporosis, or because of the spine compressing over time. What ou might not know is that everbod shrinks, and everbod also stretches ever single da! This stretching and shrinking occurs because of little discs in a person s spine. The are filled with water, acting as the bod s shock absorbers. As the da passes, these little discs lose water, compressing the spine. Then during sleep, the water is replenished, and the spine stretches back to its original size. Although elderl people do shrink, and although people epand and contract on a dail basis, people have actuall been getting taller. One proof of this observation is the height of doorwas in 1th centur homes the were not ver tall! This is because people were on average, shorter than toda. Over the last 15 ears, the average height of a person has increased b inches. Does this mean that 3 ears from now people will be an average of inches taller than what most people are now? Do ou think that the average height will eventuall reach 9 or 1 feet? Are we destined to become giants, or do ou think that this trend will stop? 119

34 Problem 1 Vertical Stretching and Compressing Now, let s eplore the effects of the A-value in the transformational function. g() 5 Af(B( C)) 1 D 1. Analze the transformational function. Where is the A-value positioned in terms of the function f: inside or outside the function? Based on the position of the A-value, do ou think it will affect the -values or the -values? Eplain our reasoning. Let s consider various A-values to understand the effects on the basic function f() 5.. Use a graphing calculator to graph each quadratic function with A.. Sketch and label the graphs. a. A $ 1 f() b., A, f() 5 1 Chapter Quadratic Functions

35 c. Consider the different A-values. How do the graphs in part (a) compare to those in part (b)? d. Does the A-value affect the -values or -values? Eplain our reasoning. A vertical dilation is a tpe of transformation that stretches or compresses an entire figure or graph. In a vertical dilation, notice that the -coordinate of ever point on the graph of a function is multiplied b a common factor, A. You can also think about this as either a vertical stretch or vertical compression. Vertical stretching is the stretching of the graph awa from the -ais. Vertical compression is the squeezing of a graph towards the -ais. 3. Is a dilation the tpe of transformation that preserves both the size and shape of a function? Eplain our reasoning.?. Consider the function f() 5 3. Dan sas that the graph of this function will look more compressed than the graph of f() 5 because the A-value is a fraction. Jeannie sas that the graph of this function will look more stretched because the A-value is greater than 1. Who is correct? Eplain our reasoning. 5. Choose a term that identifies the effect on the graph of replacing f() with Af(): vertical stretch vertical compression a. A $ 1 b., A, 1.3 Vertical Dilations of Quadratic Functions 11

36 . Given the basic function f() 5, use a graphing calculator to graph each quadratic function with A,. Sketch and label the graphs. Also, use the TABLE feature and analze each table of values. a. A # 1 f() b. 1, A, f() 5 c. How did A, affect the graph of the basic function. d. How do the graphs in part (a) compare to the graphs in Question, part (a)? e. How do the graphs in part (b) compare to the graphs in Question, part (b)? 1 Chapter Quadratic Functions

37 A reflection of a graph is a mirror image of a graph across its line of reflection. A line of reflection is the line that a graph is reflected across. 7. Identif the line of reflection for each graph in Question. Compared with the graph of 5 f(), the graph of 5 Af() is: verticall stretched b a factor of A if A. 1. verticall compressed b a factor of A if, A, 1. If A,, then the graph is also reflected across the -ais.. Given 5 f(), write the coordinate notation represented in 5 Af( C) 1 D. (, ).3 Vertical Dilations of Quadratic Functions 13

38 9. Use a graphing calculator to sketch each set of equations on the same coordinate plane. a How do the graphs of 1 and compare? b How do the graphs of 1 and 3 compare? 1. Eplain the differences between the lines of reflections used to produce and 3 in Question 9. Think about where the negative is positioned within the equation and how that affects our decision about appling the reflection. 1 Chapter Quadratic Functions

39 11. Christian, Julia, and Emil each sketched a graph of the equation 5 3 using different strategies. Provide the step-b-step reasoning used b each student. Christian A 5 1 and D 5 3 Step 1 Step 1: Step : Step Step 3: Step 3 Julia D 5 3 and A 5 1 Step 1 Step 1: Step Step : Step 3 Step 3:.3 Vertical Dilations of Quadratic Functions 15

40 Emil I rewrote the equation as 5 ( 1 3). Step 1 Step 1: Step : Step 1 Chapter Quadratic Functions

41 Given 5 f() is the basic quadratic function, ou can use reference points to graph 5 Af( C) 1 D without the use of technolog. Think about the pattern of the basic quadratic function and the A-value. Given f() 5 Graph the function g() 5 f( 3) 1 You can use reference points for f() and our knowledge about transformations to graph the function g(). From g(), ou know that A 5, C 5 3, and D 5. The verte for g() will be at (3, ). Notice A., so the graph of the function will verticall stretch b a factor of. f() 1 1 P9(3, ) f() 1 1 R9(3+, +( )) Q9(3+1, +(1 )) P9 (3, ) f() 1 1 R9(3+, +( )) g() Q9(3+1, +(1 )) P9(3, ) First, plot the new verte, (C, D). This point establishes the new set of aes. Net, think about the reference points for the basic quadratic function and that A 5. To plot point Q9 move right 1 unit and up, not 1, but 1 3 units from the verte P9 because all -coordinates are being multiplied b a factor of. To plot point R9 move right units from the P9 and up, not, but 3 units. Finall, use smmetr to complete the graph..3 Vertical Dilations of Quadratic Functions 17

42 1. Analze the worked eample. a. Use coordinate notation to represent how the A-, C-, and D-values transform the basic quadratic function to generate g() 5 f( 3) 1. (, ) b. Use the coordinate notation from part (a) to complete the table of values to verif the graph. Reference Points of Basic Quadratic Function Corresponding Points on g() (, ) (1, 1) (, ) c. Rewrite g() in terms of. 1 Chapter Quadratic Functions

43 13. Suppose function d() has the same C- and D-values the function g() in the worked eample, but its A-value is 1. a. Write d() in terms of f(). b. How would the graph of d() compare to the graph of g()? Eplain our reasoning. c. Complete the table of values. Reference Points of Basic Quadratic Function (, ) Corresponding Points on d() (, ) (1, 1) (, ) d. Sketch the graph of d(). 1 1 e. Rewrite d() in terms of..3 Vertical Dilations of Quadratic Functions 19

44 1. Suppose a function, r(), has the same C- and D-values the function g() in the worked eample, but its A-value is. a. Write r() in terms of f(). b. How would the graph of r() compare to the graph of g()? Eplain our reasoning. c. Complete the table of values. Reference Points of Basic Quadratic Function (, ) Corresponding Points on r() (, ) (1, 1) (, ) d. Sketch the graph of r(). e. Rewrite r() in terms of. 13 Chapter Quadratic Functions

45 15. Given the graph of g() 5 ( 3) 1 from the worked eample. g() a. Graph m() 5 g(). b. How is the graph of m() the same or different from the graph of r() in Question 1? c. Rewrite m() in terms of..3 Vertical Dilations of Quadratic Functions 131

46 1. Graph f() 5 1 ( 1) Write the functions that represent each graph. a. b. g() f() f() 5 g() 5 Be prepared to share our solutions and methods. 13 Chapter Quadratic Functions