Describe how to transform the parent function y = x 2 y equals , x squared  to the graph of each function below. Graph both functions on the same axes.

40. y = − 2 ( x − 1 ) 2 y equals negative 2 open x minus 1 , close squared
41. y = − 2 ( x + 1 ) 2 + 1 y equals negative 2 open x plus 1 , close squared , plus 1
42. y = 3 ( x − 2 ) 2 + 3 y equals 3 open x minus 2 , close squared , plus 3
43. y = − 1 ( x + 4 ) 2 + 5 y equals negative 1 open x plus 4 , close squared , plus 5
44. y = − 0 . 25 x 2 + 3 y equals negative 0 . 25 , x squared , plus 3
45. y = 0 . 2 ( x − 12 ) 2 − 3 y equals 0 . 2 open x minus 12 , close squared , minus 3

46. Error Analysis A classmate graphed y = − 2 ( x − 3 ) 2 + 4 y equals negative 2 open x minus 3 , close squared , plus 4  as shown. Explain your classmate's error. Graph the correct parabola.

    

47. Writing Describe the family of quadratic functions whose members each have (3, 4) as their vertex.
48. Write a quadratic function to represent the areas of all rectangles with a perimeter of 36 ft. Graph the function and describe the rectangle that has the largest area.

Write the equation of each parabola in vertex form.

49. vertex (1, 2), point ( 2 , − 5 ) open 2 comma negative 5 close
50. vertex ( − 3 , 6 ) , open negative 3 comma 6 close comma  point ( 1 , − 2 ) open 1 comma negative 2 close
51. vertex (0, 5), point ( 1 , − 2 ) open 1 comma negative 2 close
52. vertex ( 1 4 , − 3 2 ) , open . 1 fourth , comma negative , 3 halves . close . comma  point (1, 3)

In Chapter 2, you graphed absolute value functions as transformations of their parent function y = | x | . y equals vertical line x vertical line .  Similarly, you can graph a quadratic function as a transformation of the parent function y = x 2 . y equals , x squared , .  Graph the following pairs of functions on the same set of axes. Determine how they are similar and how they are different.

53. y = − | x − 2 | + 1 ; y = − ( x − 2 ) 2 + 1 y equals negative vertical line x minus 2 vertical line plus 1 semicolon y equals negative open x minus 2 , close squared , plus 1
54. y = 3 | x + 1 | − 2 ; y = 3 ( x + 1 ) 2 − 2 y equals 3 vertical line x plus 1 vertical line negative 2 semicolon y equals 3 open x plus 1 , close squared , minus 2
55. y = − 2 | x | + 4 ; y = − 2 x 2 + 4 y equals negative 2 vertical line x vertical line plus 4 semicolon y equals negative 2 , x squared , plus 4
56. y = | x + 3 | ; y = ( x + 3 ) 2 y equals vertical line x plus 3 vertical line semicolon y equals open x plus 3 , close squared
57. Open-Ended Write an equation of a parabola symmetric about x = − 10 . x equals negative 10 .
58. 1. Technology Determine the axis of symmetry for each parabola defined by the spreadsheet values below.

       	A	B
       1	X1	Y1
       2	1	− 35 negative 35
       3	2	− 15 negative 15
       4	3	− 3 negative 3
       5	4	1
       6	5	− 3 negative 3

       	A	B
       1	X2	Y2
       2	1	10
       3	2	2
       4	3	2
       5	4	10
       6	5	26

    2. How could you use the spreadsheet columns to verify that the axes of symmetry are correct?

    3. What functions in vertex form model the data?

       Check that the axes of symmetry are correct.

C Challenge

Determine a and k so the given points are on the graph of the function.

59. (0, 1), (2, 1); y = a ( x − 1 ) 2 + k y equals eh open x minus 1 , close squared , plus k
60. ( − 3 , 2 ) , open negative 3 comma 2 close comma  (0, 11); y = a ( x + 2 ) 2 + k y equals eh open x plus 2 , close squared , plus k
61. (1, 11), ( 2 , − 19 ) ; y = a ( x + 1 ) 2 + k open 2 comma negative 19 close semicolon y equals eh open x plus 1 , close squared , plus k
62. ( − 2 , 6 ) , open negative 2 comma 6 close comma  (3, 1); y = a ( x − 3 ) 2 + k y equals eh open x minus 3 , close squared , plus k
63. 1. In the function y = a x 2 + b x + c , y equals eh , x squared , plus b x plus c comma  c represents the y-intercept. Find the value of the y-intercept in the function y = a ( x − h ) 2 + k . y equals eh . open x minus h close squared . plus k .
    2. Under what conditions does k represent the y-intercept?

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