3.1. Shape and Structure. Forms of Quadratic Functions 169A

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1 Shape and Structure Forms of Quadratic Functions.1 Learning goals Ke Terms 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. standard form of a quadratic function factored form of a quadratic function vertex form of a quadratic function concavit of a parabola Essential Ideas The standard form of a quadratic function is written as f(x) 5 ax 2 1 bx 1 c, where a does not equal 0. The factored form of a quadratic function is written as f(x) 5 a(x 2 r 1 ) (x 2 r 2 ), where a does not equal 0. The vertex form of a quadratic function is written as f(x) 5 a(x 2 h) 2 1 k, where a does not equal 0. The concavit of a parabola describes whether a parabola opens up or opens down. A parabola is concave down if it opens downward, and is concave up if it opens upward. Texas Essential Knowledge and Skills for MathematiCS (4) Quadratic and square root functions, equations, and inequalities. The student applies mathematical processes to understand that quadratic and square root functions, equations, and quadratic inequalities used to model situations, solve problems, and make predictions. The student is expected to: (B) write the equation of a parabola using given attributes, including vertex, focus, directrix, axis of smmetr, and direction of opening (D) transform a quadratic function f(x) 5 a x 2 1 bx 1 c to the form f(x) 5 a(x 2 h ) 2 1 k to identif the different attributes of f(x) 169A

2 Overview Students will match quadratic equations with their graphs using ke characteristics. The standard form, the factored form, and the vertex form of a quadratic equation are reviewed as is the concavit of a parabola. Students then paste each of the functions with their graphs into one of three tables depending on the form in which the equation is written while identifing ke characteristics of each function such as the axis of smmetr, the x-intercept(s), concavit, the vertex, and the -intercepts. Next, a graph is presented on a numberless coordinate plane and students identif which function(s) could model it based on its ke characteristics. Finall, a worked example shows that a unique quadratic function is determined when the vertex and a point on the parabola are known, or the roots and a point on the parabola are known. Students are given information about a function, and the will determine the most efficient form (standard, factored, vertex) to write the function, based on the given information. 169B Chapter Quadratic Functions

3 Warm Up 1. Consider the quadratic functions shown. A. f(x) 5 5x 2 1 4x 1 10 B. g(x) 5 5(x 2 4) (x 1 ) C. h(x) 5 5(x 2 ) a. Which function is written in vertex form? How do ou know? h(x) 5 5(x 2 ) is written in vertex form, because it is written in f(x) 5 a(x 2 h) 2 1 k form. b. Which function is written in standard form? How do ou know? f(x) 5 5x 2 1 4x 1 10 is written in standard form, because it is written in f(x) 5 ax 2 1 bx 1 c form. c. Which function is written in factored form? How do ou know? g(x) 5 5(x 2 4)(x 1 ) is written in factored form, because it is written in f(x) 5 a(x 2 r 1 )(x 1 r 2 ) form..1 Forms of Quadratic Functions 169C

4 169D Chapter Quadratic Functions

5 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. KeY TeRmS standard form of a quadratic function factored form of a quadratic function vertex 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 1974 with the help of some friends, he spent all night secretl placing a 450 pound cable between the World Trade Center Towers in New York Cit. At dawn, to the shock and amazement of onlookers, the fatigued 24-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. Have ou ever challenged ourself to do something difficult just to see if ou could do it? You can see the events unfold in the 2002 Academ Award winning documentar Man on Wire b James Marsh Forms of Quadratic Functions 169

6 Problem 1 Students will match nine quadratic functions with their appropriate graphs using the vertex, x-intercepts, -intercept, and a-value, depending on the form of the quadratic function. The standard form, factored form, and vertex form of a quadratic equation are reviewed and students then sort their functions and graphs into groups based on these forms. The concavit of a parabola is reviewed, and students identif the ke characteristics that can be determined from a quadratic equation written in each form. Students then paste each of the functions and their graphs into one of three tables, depending on the form in which the equation is written. The also identif the axis of smmetr, the x-intercept(s), concavit, the vertex, and the -intercepts for each function. Problem 1 It s All in the Form 1. Cut out each quadratic function and graph on the next page two pages. a. Tape each quadratic function to its corresponding graph. Graph A, Function b Graph F, Function e Graph B, Function a Graph C, Function h Graph D, Function f Graph E, Function c Graph G, Function i Graph H, Function g Graph I, Function d 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. Explain the method(s) ou used to match the functions with their graphs. Answers will var. Students ma identif the graphs b their vertex, x-intercept(s), -intercept, and a-value depending on the form of the quadratic function. The ma also substitute values of points into the functions or make a table. Grouping Have students complete Question 1 with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Question 1 What feature of the quadratic function helps determine if the parabola opens up or down? What feature of the quadratic function helps determine if it has a maximum or a minimum? 170 Chapter Quadratic Functions What feature of the quadratic function helps determine if the parabola passes through the origin? What feature of the quadratic function helps determine the -intercept? What feature of the quadratic function helps determine the x-intercept(s)? What feature of the quadratic function helps determine the location of the vertex? 170 Chapter Quadratic Functions

7 a. f(x) 5 2(x 1 1)(x 1 5) d. f(x) 5 (x 2 1) 2 g. f(x) 5 2(x 1 4) b. f(x) 5 1 x2 1 πx e. f(x) 5 2(x 2 1)(x 2 5) h. f(x) 5 25x 2 2x 1 21 c. f(x) (x 2 )(x 2 ) f. f(x) 5 x x 2 1 i. f(x) 5 2(x 1 2) Forms of Quadratic Functions Forms of Quadratic Functions 171

8 172 Chapter Quadratic Functions 172 Chapter Quadratic Functions

9 A. B. C x x x tape function here b. f(x) 5 1 x2 1 πx tape function here a. f(x) 5 2(x 1 1)(x 1 5) tape function here h. f(x) 5 25x 2 2 x 1 21 D. E. F x x x tape function here tape function here tape function here f. f(x) 5 x x 2 1 c. f(x) (x 2 )(x 2 ) e. f(x) 5 2(x 2 1)(x 2 5) G. H. I x x x tape function here i. f(x) 5 2(x 1 2) tape function here g. f(x) 5 2(x 1 4) tape function here d. f(x) 5 (x 2 1) 2.1 Forms of Quadratic Functions 17.1 Forms of Quadratic Functions 17

10 174 Chapter Quadratic Functions 174 Chapter Quadratic Functions

11 Grouping Ask a student to read the information. Discuss as a class. Have students complete Questions 2 and with a partner. Then have students share their responses as a class. Recall that quadratic functions written in different forms. standard form: f(x) 5 ax 2 1 bx 1 c, where a does not equal 0. factored form: f(x) 5 a(x 2 r 1 )(x 2 r 2 ), where a does not equal 0. vertex form: f(x) 5 a(x 2 h) 2 1 k, where a does not equal Sort our graphs with matching equations into piles based on the function form. Keep these piles; ou will use them again at the end of this Problem. Make a list of words used to describe quadratic functions: x-intercept, -intercept, vertex, axis of smmetr, and concave up or down. Pronounce each word aloud, having students repeat after ou. Draw a quadratic function graph on the board. Use the graph to help define each vocabular word. The graphs of quadratic functions described using ke characteristics: x-intercept(s), -intercept, vertex, axis 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.. The form of a quadratic function highlights different ke characteristics. State the characteristics ou can determine from each. standard form I can determine the -intercept, and whether the parabola is concave up or down when the quadratic is in standard form. factored form Guiding Questions for Share Phase, Questions 2 and Which form(s) of the quadratic function is used to easil identif the -intercept? Which form(s) of the quadratic function is used to easil identif if the parabola opens up or down? Which form(s) of the quadratic function is used to easil identif the x-intercept(s)? I can determine the x-intercepts, and whether the parabola is concave up or concave down when the quadratic is in factored form. vertex form I can determine the vertex, whether the parabola is concave up or concave down, and the axis of smmetr when the quadratic is in vertex form..1 Forms of Quadratic Functions 175 Which form(s) of the quadratic function is used to easil identif the location of the vertex? Which form(s) of the quadratic function is used to easil identif the axis of smmetr?.1 Forms of Quadratic Functions 175

12 Grouping Ask a student to read the information and student work. Complete Question 4 as a class. 4. Christine, Kate, and Hannah were asked to determine the vertex of three different quadratic functions each written in different forms. Analze their calculations. Christine f(x) 5 2x x 1 10 The quadratic function is in standard form. So I know the axis of smmetr is x 5 2b 2a ẋ 5 2(2) Now that I know the axis of smmetr, I can substitute that value into the function to determine the -coordinate of the vertex. f(2) 5 2(2) (2) (9) Therefore, the vertex is (2, 8). Kate g(x) 5 1_ (x 1 )(x 2 7) 2 The form of the function tells me the x-intercepts are 2 and 7. I also know the x-coordinate of the vertex will be directl in the middle of the x-intercepts. So, all I have to do is calculate the average. x Now that I know the x-coordinate of the vertex, I can substitute that value into the function to determine the -coordinate. g(2) 5 1_ (2 1 )(2 2 7) 2 5 1_ 2 (5)(25) Therefore, the vertex is (2, 212.5). Hannah h(x) 5 2 x x 1 17 I can determine the vertex b rewriting the function in vertex form.to do that, I need to complete the square. h(x) 5 2 x x ( x 2 1 6x 1 ) ( x 2 1 6x 1 9) (x 1 ) Now, I know the vertex is (2, 21). 176 Chapter Quadratic Functions 176 Chapter Quadratic Functions

13 a. How are these methods similar? How are the different? Christine s method and Kate s method require that ou determine the axis of smmetr, and then substitute that value into the function to determine the -coordinate of the vertex. Their methods are different in the wa the axis of smmetr was determined. Christine used x 5 2b 2a and Kate used x 5 r 1 r Hannah completed the square to rewrite her equation in vertex form. When a quadratic equation is in vertex form, f(x) 5 a(x 2 h ) 2 1 k, the coordinates of the vertex are (h, k). b. What must Kate do to use Christine s method? Kate knows the a-value from the form of her quadratic equation. She must multipl the factors together and combine like terms. She would then have a quadratic function in standard form to determine the b-value. c. What must Christine do to use Kate s method? Christine must factor the quadratic function or use the quadratic formula to determine the x-intercepts. Once she determines the x-intercepts, she can use the same method as Kate. d. Describe the steps Hannah used to complete the square and rewrite her equation in vertex form. To complete the square and rewrite her equation in vertex form, Hannah completed the following steps: Factor out a 2 from 2 x x. Complete the square b adding 9 to x 2 1 6x. She calculated 9 b dividing the coefficient of 6x b 2, then squaring the result, ( 6 2 ) 2. Add 218 to maintain balance in the equation. Adding 9 to x 2 1 6x results in adding 18 to the equation because the quantit ( x 2 1 6x 1 9) is multiplied b 2. Adding 218 maintains balance in the equation. Rewrite x 2 1 6x 1 9 in factored form, (x 1 ) 2, and subtract Forms of Quadratic Functions Forms of Quadratic Functions 177

14 Grouping Have students complete Question 5 with a partner. Then have students share their responses as a class. 5. Analze each table on the following three pages. Paste each function and its corresponding graph from Question 2 in the Graphs and Their Functions section of the appropriate table. Then, explain how ou can determine each ke characteristic based on the form of the given function. Guiding Questions for Share Phase, Question 5 Which ke characteristics are observable when the quadratic function is written in standard form? Which ke characteristics are observable when the quadratic function is written in factored form? Which ke characteristics are observable when the quadratic function is written in vertex form? Which formulas are associated with determining the ke characteristics of a quadratic function written in standard form? Which formulas are associated with determining the ke characteristics of a quadratic function written in factored form? Which formulas are associated with determining the ke characteristics of a quadratic function written in vertex form? A b. f(x) 5 1 x2 1 πx x 178 Chapter Quadratic Functions Standard Form f(x) 5 ax 2 1 bx 1 c, where a fi 0 C. Graphs and Their Functions h. f(x) 5 25x 2 2 x 1 21 Methods to Identif and Determine Ke Characteristics 1 2 x D f. f(x) 5 x x 2 1 Axis of Smmetr x-intercept(s) Concavit x 5 2b 2a Vertex Use 2b to determine the x-coordinate of the 2a vertex. Then substitute that value into the equation and solve for. Substitute 0 for, and then solve for x using the quadratic formula, factoring, or a graphing calculator. Concave up when a. 0 Concave down when a, 0 -intercept c-value x 178 Chapter Quadratic Functions

15 Factored Form f(x) 5 a(x 2 r 1 )(x 2 r 2 ), where a fi 0 Graphs and Their Functions B. E. F x x x a. f(x) 5 2(x 1 1)(x 1 5) c. f(x) (x 2 )(x 2 ) e. f(x) 5 2(x 2 1)(x 2 5) Methods to Identif and Determine Ke Characteristics Axis of Smmetr x-intercept(s) Concavit x 5 r 1 r (r 1, 0), (r 2, 0) Concave up when a. 0 Concave down when a, 0 Vertex Use r 1 r 1 2 to determine the x-coordinate of the 2 vertex. Then substitute that value into the equation and solve for. -intercept Substitute 0 for x, and then solve for..1 Forms of Quadratic Functions Forms of Quadratic Functions 179

16 Vertex Form f(x) 5 a(x 2 h) 2 1 k, where a fi 0 Graphs and Their Functions G. H. I x x x i. f(x) 5 2(x 1 2) g. f(x) 5 2(x 1 4) d. f(x) 5 (x 2 1) 2 Methods to Identif and Determine Ke Characteristics Axis of Smmetr x-intercept(s) Concavit x 5 h Substitute 0 for, and then solve for x using the quadratic formula, factoring, or a graphing calculator. Concave up when a. 0 Concave down when a, 0 Vertex -intercept (h, k) Substitute 0 for x, and then solve for. 180 Chapter Quadratic Functions 180 Chapter Quadratic Functions

17 Problem 2 Given a graph on a numberless coordinate plane, students will identif functions that model the graph. Next, the identif the form of the function, and rewrite the function in the other two forms when possible. Problem 2 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. x Grouping Have students complete Questions 1 and 2 with a partner. Then have students share their responses as a class. f 1 (x) 5 22(x 1 1)(x 1 4) The function f 1 is a possibilit because it has a negative a-value and 2 negative x-intercepts. f 2 (x) x2 2 x 2 6 f (x) 5 2(x 1 1)(x 1 4) The function f 2 is a possibilit because it has a negative a-value and a negative -intercept. The function f eliminated because it has a positive a-value which means the graph would be concave up. Guiding Questions for Share Phase, Question 1 part (a) Are the zeros of the quadratic function real or imaginar? How do ou know? Are the zeros of the quadratic function negative or positive or both? How do ou know? Is the a-value of the quadratic function negative or positive? How do ou know? How man of the functions have a vertex in the second quadrant? How man of the functions have 2 negative x-intercepts? How man of the functions have a negative -intercept? Think about the information given b each function and the relative position of the graph. f 4 (x) 5 2x f 5 (x) 5 2(x 2 1)(x 2 4) f 6 (x) 5 2(x 2 6) 2 1 The function f 4 eliminated because it has a positive a-value which means the graph would be concave up. The function f 5 eliminated because it has a positive a-value which means the graph would be concave up. f 7 (x) 5 2(x 1 2)(x 2 ) f 8 (x) 5 2(x 1 6) 2 1 The function f 7 eliminated because it has one positive and one negative x-intercept. The function f 8 is a possibilit because it has a negative a-value and a vertex in Quadrant II. The function f 6 eliminated because its vertex is in Quadrant I..1 Forms of Quadratic Functions Forms of Quadratic Functions 181

18 Guiding Questions for Share Phase, Question 1 part (b) Are the zeros of the quadratic function real or imaginar? How do ou know? Are the zeros of the quadratic function negative or positive or both? How do ou know? Is the a-value of the quadratic function negative or positive? How do ou know? How man of the functions have a vertex in the fourth quadrant? How man of the functions have 2 positive x-intercepts? How man of the functions have a positive -intercept? b. f 1 (x) 5 2(x 2 75) f 2 (x) 5 (x 2 8)(x 1 2) f (x) 5 8x x The function f 1 is a possibilit because it has a positive a-value making it concave up, and a vertex in Quadrant IV. The function f 2 eliminated because it does not have 2 positive x-intercepts. x The function f is a possibilit because it has a positive a-value making it concave up, and a positive -intercept. f 4 (x) 5 2(x 2 1)(x 2 5) f 5 (x) 5 22(x 2 75) f 6 (x) 5 x 2 1 6x 2 2 The function f 4 eliminated because it has a negative a-value which means the graph would be concave down. The function f 5 eliminated because it has a negative a-value which means the graph would be concave down. The function f 6 eliminated because it has a negative -intercept. f 7 (x) 5 2(x 1 4) f 8 (x) 5 (x 1 1)(x 1 ) The function f 7 eliminated because it has a vertex in Quadrant III. The function f 8 eliminated because it has 2 negative x-intercepts. 182 Chapter Quadratic Functions 182 Chapter Quadratic Functions

19 Guiding Questions for Share Phase, Question 1 part (c) Are the zeros of the quadratic function real or imaginar? How do ou know? Are the zeros of the quadratic function negative or positive or both? How do ou know? Is the a-value of the quadratic function negative or positive? How do ou know? How man of the functions have a vertex in the first quadrant? How man of the functions have no x-intercepts? How man of the functions have a positive -intercept? c. f 1 (x) 5 (x 1 1)(x 2 5) f 2 (x) 5 2(x 1 6) f (x) 5 4x x 1 10,010 The function f 1 eliminated because it has real x-intercepts. The function f 2 eliminated because it has a vertex in Quadrant III. x The function f is a possibilit because it has a positive -intercept and a positive a-value. f 4 (x) 5 (x 1 1)(x 1 5) f 5 (x) 5 2(x 2 6) f 6 (x) 5 x 2 1 2x 2 5 The function f 4 eliminated because it has real x-intercepts. The function f 5 is a possibilit because it has a positive a-value and a vertex in Quadrant I. The function f 6 eliminated because it has a negative -intercept..1 Forms of Quadratic Functions 18.1 Forms of Quadratic Functions 18

20 Guiding Questions for Share Phase, Question 2 What algebraic properties are used to change the quadratic function in part (a) from factored form to standard form? What algebraic properties are used to change the quadratic function in part (a) from factored form to vertex form? What algebraic properties are used to change the quadratic function in part (b) from standard form to vertex form? Wh can t the quadratic function in part (b) be written in factored form? What algebraic properties are used to change the quadratic function in part (c) from vertex form to standard form? Wh can t the quadratic function in part (c) be written in factored form? 2. Consider the three functions shown from Question 1. Identif the form of the function given. Write the function in the other two forms, if possible. If it is not possible, explain wh. Determine the -intercept, x-intercepts, axis of smmetr, vertex, and concavit. a. From part (a): f 1 (x) 5 22(x 1 1)(x 1 4) The function is given in factored form. Standard Form: Vertex Form: f 1 (x) 5 22(x 1 1)(x 1 4) f 1 (x) 5 22(x 2 1 5x 1 4) 5 22(x 2 1 5x 1 4) 5 22 ( x 2 1 5x ) x x ( 5 x 1 2 ) The -intercept is (0, 28). In standard form, f(x) 5 ax 2 1 bx 1 c, c represents the -intercept. The x- intercepts are (24, 0) and (21, 0). In factored form, f(x) 5 a(x 2 r 1 )(x 2 r 2 ), r 1 and r 2 represent the x-intercepts. 5 The axis of smmetr is x In vertex form, f(x) 5 a(x 2 h)2 1 k, h represents the axis of smmetr. The vertex ( , 2 ). In vertex form, f(x) 5 a(x 2 h)2 1 k, (h, k) represents the vertex. The parabola is concave up because the value of a is positive in vertex form f(x) 5 a(x 2 h) 2 1 k. b. From part (c): f (x) 5 4x x 1 10,010 The function is given in standard form. Vertex Form: Factored Form: f (x) 5 4x x 1 10,010 Answers will var. 5 4(x x ) 1 10, ,000 The function does not cross the x-axis, therefore it does not have 5 4(x 2 50) real number x-intercepts. I cannot factor this function. The - intercept is (0, 10,010). In standard form, f(x) 5 ax 2 1 bx 1 c, c represents the - intercept. There are no real x- intercepts because I know the function does not cross the x-axis. The axis of smmetr is x In vertex form, f(x) 5 a(x 2 h) 2 1 k, h represents the axis of smmetr. The vertex is (50, 10). In vertex form, f(x) 5 a(x 2 h) 2 1 k, (h, k) represents the vertex. The parabola is concave up because the value of a is positive in vertex form f(x) 5 a(x 2 h) 2 1 k. 184 Chapter Quadratic Functions 184 Chapter Quadratic Functions

21 c. From part (c): f 5 (x) 5 2(x 2 6) The function is given in vertex form. Standard Form: f 5 (x) 5 2_x Factored Form: Answers will var. 5 2_x x The function does not cross the x-axis, therefore it does not have 5 2x x real number x-intercepts. I cannot 5 2x x 1 77 factor this function. The - intercept is (0, 77). In standard form, f(x) 5 ax 2 1 bx 1 c, c represents the - intercept. There are no real x- intercepts because I know the function does not cross the x-axis. The axis of smmetr is x 5 6. In vertex form, f(x) 5 a(x 2 h) 2 1 k, h represents the axis of smmetr. The vertex is (6, 5). In vertex form, f(x) 5 a(x 2 h) 2 1 k, (h, k) represents the vertex. The parabola is concave up because the value of a is positive in vertex form f(x) 5 a(x 2 h) 2 1 k..1 Forms of Quadratic Functions Forms of Quadratic Functions 185

22 Problem Students will explore the number of unknowns when quadratic functions are written in the different forms. A worked example shows that a unique quadratic function is determined when the vertex and a point on the parabola are known, or the roots and a point on the parabola are known. In the last activit, students are given information about the function, and the then determine the most efficient form (standard, factored, vertex) to write the function, based on the given information. Grouping Have students complete Question 1 on their own and discuss. Complete Question 2 as a class. Problem Unique... one and onl? 1. George and Pat were each asked to write a quadratic equation with a vertex of (4, 8). Analze each student s work. Describe the similarities and differences in their equations and determine who is correct. George 5 a(x 2 h) 2 1 k 5 a(x 2 4) _ 2 (x 2 4)2 1 8 Pat 5 a(x 2 h) 2 1 k 5 a(x 2 4) (x 2 4) Both George and Pat are correct. George and Pat each used the vertex form of a quadratic 1 equation and substituted h 5 4 and k 5 8. George chose a 5 2 and Pat chose a There was not information given to create a unique quadratic equation, therefore, both equations represent a quadratic equation with the vertex (4, 8). 2. Consider the forms of quadratic functions and state the number of unknown values in each. Form Number of Unknown Values f(x) 5 a(x 2 h) 2 1 k 5 f(x) 5 a(x 2 r 1 )(x 2 r 2 ) 5 f(x) 5 ax 2 1 bx 1 c 5 a. If a function is written in vertex form and ou know the vertex, what is still unknown? I still have unknowns: x,, and a. Guiding Questions for Discuss Phase, Questions 1 and 2 part (c) George wrote his quadratic equation using which form? Pat wrote her quadratic equation using which form? Do both quadratic equations have a vertex at (4, 8)? How man different parabolas could have a vertex at (4, 8)? If a quadratic function is written in vertex form, and the vertex is given, which variables are known? 186 Chapter Quadratic Functions b. If a function is written in factored form and ou know the roots, what is still unknown? I still have unknowns: x,, and a. 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. If the function is written in vertex form, I still have unknowns: a, h, and k. If the function is written in factored form, I still have unknowns: r 1, r 2, and a. If the function is written in standard form, I still have unknowns: a, b, and c. If a quadratic function is written in factored form, and the roots are given, which variables are known? How man different quadratic functions have the same vertex? How man different quadratic functions have the same zeros? Will knowing the vertex and the zeros determine a unique quadratic function? Will knowing the vertex and the a-value determine a unique quadratic function? 186 Chapter Quadratic Functions

23 Guiding Questions for Discuss Phase, Question 2 parts (d) and (e) Will knowing the vertex and a point on the function determine a unique quadratic function? Will knowing the roots and a point on the function determine a unique quadratic function? Will knowing the roots and the a-value on the function determine a unique quadratic function? Grouping Ask a student to read the information and the worked examples. Complete Questions and 4 as a class. d. If ou onl know the vertex, what more do ou need to write a unique function? Explain our reasoning. I will need a point or the a-value. If I have a point I can solve for the a-value, if I have the a-value, then I have the unique equation. e. If ou onl know the roots, what more do ou need to write a unique function? Explain our reasoning. I will need a point or the a-value. If I have a point I can solve for the a-value, if I have the a-value, then I have the unique equation. You can write a unique quadratic function given a vertex and a point on the parabola. Write the quadratic function given the vertex (5, 2) and the point (4, 9). Substitute the given values into the vertex form of the function. Then simplif. Finall, substitute the a-value into the function. f(x) 5 a(x 2 h) 2 1 k 9 5 a(4 2 5) a(21) a a 7 5 a f(x) 5 7(x 2 5) You can write a unique quadratic function given the roots and a point on the parabola. Write a quadratic function given the roots (22, 0) and (4, 0), and the point (1, 6). Substitute the given values into the factored form of the function. Then simplif. Finall, substitute the a-value into the function. f(x) 5 a(x 2 r 1 )(x 2 r 2 ) 6 5 a(1 2 (22))(1 2 4) 6 5 a(1 1 2)(1 2 4) 6 5 a()(2) a a f(x) (x 1 2)(x 2 4) _A2_TX_Ch0_ indd Forms of Quadratic Functions /0/14 2:2 PM.1 Forms of Quadratic Functions 187

24 . Explain wh knowing the vertex and a point creates a unique quadratic function. A unique quadratic function is created because 4 of the 5 unknowns are given, which means there is onl one possible a-value. 4. If ou are given the roots, how man unique quadratic functions can ou write? Explain our reasoning. I can write an infinite number of quadratic functions. If I am onl given the roots, I can assign an a-value that I want. Grouping Have students complete Question 5 with a partner. Then have students share their responses as a class. 5. Use the given information to determine the most efficient form ou could use to write the function. Write standard form, factored form, vertex form, or none in the space provided. a. minimum point (6, 275) vertex form -intercept (0, 15) b. points (2, 0), (8, 0), and (4, 6) factored form Guiding Questions for Share Phase, Question 5 If ou know the minimum or maximum point of a quadratic function, is that alwas the vertex of the function? Under what circumstances is it easier to write the function in vertex form? If ou know three points on the quadratic function, is it alwas easier to write the function in factored form? Under what circumstances is it easier to write the function in factored form? Under what circumstances is it easier to write the function in standard form? Is the maximum height of Max s baseball the location of the vertex of the parabola? 188 Chapter Quadratic Functions c. points (100, 75), (450, 75), and (150, 95) standard form d. points (, ), (4, ), and (5, ) none e. x-intercepts: (7.9, 0) and (27.9, 0) factored form point (24, 24) f. roots: (, 0) and (12, 0) factored form point (10, 2) g. Max hits a baseball off a tee that is feet high. vertex form The ball reaches a maximum height of 20 feet when it is 15 feet from the tee. h. A grasshopper was standing on the 5 ard factored form line of a football field. He jumped, and landed on the 8 ard line. At the 6 ard line he was 8 inches in the air. Are the 5 and 8 ard lines on the football field the location of the roots of the parabola? 188 Chapter Quadratic Functions

25 Problem 4 Students will write a quadratic function to represent each of three given situations. Grouping Have students complete Questions 1 through with a partner. Then have students share their responses as a class. Guiding Questions for Share Phase, Questions 1 through Which ke characteristic is associated with the given information? If Amazing Larr reaches a maximum height of 0 feet, will this be the vertex of the graph? Did ou write the quadratic function in vertex form? If the cannon is 10 feet above the ground, is this associated with the -intercept of the graph? What is the concavit of the parabola? How do ou know? Is the a-value of the function a positive number or a negative number? What distance has Craz Cornelius traveled before he leaves the ground? What distance has Craz Cornelius traveled as he lands back on the ground? Are the points or distances at which Craz Cornelius lifts off the ground and lands back down on the ground associated with the roots of the function or x-intercepts? Problem 4 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 maximum height of 0 feet during his next launch. Based on Amazing Larr s previous launches, his assistant DaJuan has estimated that this will occur when he is 40 feet from the cannon. When Amazing Larr is shot from the cannon, he is 10 feet above the ground. Write a function to represent Amazing Larr s height in terms of his distance. Let h(d) represent Amazing Larr s height in terms of his distance, d. h(d) 5 a(d 2 40) a(0 2 40) a a a h(d) (d 2 40) Craz Cornelius is a fire jumper. He is attempting to run and jump through a ring of fire. He runs for 10 feet. Then, he begins his jump just 4 feet from the fire and lands on the other side feet from the fire ring. When Cornelius was 1 foot from the fire ring at the beginning of his jump, he was.5 feet in the air. Write a function to represent Craz Cornelius height in terms of his distance. Round to the nearest hundredth. Let h(d) represent Craz Cornelius s height in terms of his distance, d. h(d) 5 a(d 2 r 1 )(d 2 r 2 ).5 5 a(1 2 10)(1 2 17).5 5 a()(24) a a h(d) (d 2 10)(d 2 17). Harsh Knarsh is attempting to jump across an alligator filled swamp. She takes off from a ramp 0 feet high with a speed of 95 feet per second. Write a function to represent Harsh Knarsh s height in terms of time. Let h(t) represent Harsh Knarsh s height in terms of his time, t. h(t) 5 216t 2 1 v 0 t 1 h 0 h(t) 5 216t t 1 0 Be prepared to share our solutions and methods. Remember, the general equation to represent height over time is h(t) 5 216t 2 1 v 0 t 1 h 0 where v 0 is the initial velocit in feet per second and h 0 is the initial height in feet..1 Forms of Quadratic Functions 189 What distance has Craz Cornelius traveled when he reaches a height of.5 feet? Did ou write the quadratic function in factored form? If Harsh Knarsh begins on a ramp 0 feet high, is this associated to the -intercept of the graph? What function or equation is associated with vertical motion? Is the equation for vertical motion written in standard form?.1 Forms of Quadratic Functions 189

26 Check for Students Understanding 1. A catapult is designed to hurl a pumpkin from a height of 40 feet at an initial velocit of 215 feet per second. a. Write a quadratic function to represent the situation using the given information. h(t) 5 216t t 1 40 b. Write an equation that ou can use to determine when the pumpkin will hit the ground. Then solve the equation. h(t) 5 216t t t t 1 40 a5 216 b5 215 c5 40 t (216)(40) 2(216) , , t ,785 t , < <1.62 The pumpkin will hit the ground after about 1.62 seconds. 190 Chapter Quadratic Functions