1. Splash Screen

2. Key Concept: Linear and Polynomial Parent Functions
Five-Minute Check
Then/Now
New Vocabulary
Key Concept: Linear and Polynomial Parent Functions
Key Concept: Square Root and Reciprocal Parent Functions
Key Concept: Absolute Value Parent Function
Key Concept: Greatest Integer Parent Function
Example 1: Describe Characteristics of a Parent Function
Key Concept: Vertical and Horizontal Translations
Example 2: Graph Translations
Key Concept: Reflections in the Coordinate Axes
Example 3: Write Equations for Transformations
Key Concept: Vertical and Horizontal Dilations
Example 4: Describe and Graph Transformations
Example 5: Graph a Piecewise-Defined Function
Example 6: Real-World Example: Transformations of Functions
Key Concept: Transformations with Absolute Value
Example 7: Describe and Graph Transformations
Lesson Menu

3. Estimate and classify the extrema for f (x)
Estimate and classify the extrema for f (x). Support your answer numerically.
5–Minute Check 1

4. A. Relative minimum: (–1.5, –9.375); relative maximum: (1, 0)
B. Relative minimum: (–0.5, –4.5); relative maximum: (1, 0)
C. Relative minimum: (–1.5, –9.375); relative maximum: (0.5, 0)
D. Relative minimum: (–2, –9); relative maximum: (0.5, 0)
5–Minute Check 1

5. A. Relative minimum: (–1.5, –9.375); relative maximum: (1, 0)
B. Relative minimum: (–0.5, –4.5); relative maximum: (1, 0)
C. Relative minimum: (–1.5, –9.375); relative maximum: (0.5, 0)
D. Relative minimum: (–2, –9); relative maximum: (0.5, 0)
5–Minute Check 1

6. Estimate and classify the extrema for f (x)
Estimate and classify the extrema for f (x). Support your answers numerically.
5–Minute Check 2

7. A. Absolute minimum: (–2, –4) B. Absolute minimum: (–2, –8)
C. Relative minimum: (–2, –4)
D. Absolute minimum: (–1, –4)
5–Minute Check 2

8. A. Absolute minimum: (–2, –4) B. Absolute minimum: (–2, –8)
C. Relative minimum: (–2, –4)
D. Absolute minimum: (–1, –4)
5–Minute Check 2

9. Find the average rate of change of f (x) = 3x 3 – x 2 + 5x – 3 on the interval [–1, 2].
B. 5 C.
D. 13
5–Minute Check 3

10. Find the average rate of change of f (x) = 3x 3 – x 2 + 5x – 3 on the interval [–1, 2].
B. 5 C.
D. 13
5–Minute Check 3

11. You analyzed graphs of functions. (Lessons 1-2 through 1-4)
Identify, graph, and describe parent functions.
Identify and graph transformations of parent functions
Then/Now

12. absolute value function step function greatest integer function
parent function
constant function
zero function
identity function
quadratic function
cubic function
square root function
reciprocal function
absolute value function
step function
greatest integer function
transformation
translation
reflection
dilation
Vocabulary

13. Key Concept 1

14. Key Concept 2

15. Key Concept 3

16. Key Concept 4

17. Describe Characteristics of a Parent Function
Describe the following characteristics of the graph of the parent function : domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.
Example 1

18. The domain of the function is, and the range is .
Describe Characteristics of a Parent Function
The graph of the reciprocal function shown in the graph has the following characteristics.
The domain of the function is, and the range is
The graph has no intercepts.
The graph is symmetric with respect to the origin, so f (x) is odd.
Example 1

19. Describe Characteristics of a Parent Function
The graph is continuous for all values in its domain with an infinite discontinuity at x = 0.
The end behavior is as
The graph is decreasing on the interval and decreasing on the interval
Example 1

20. Describe Characteristics of a Parent Function
Answer:
Example 1

21. Describe Characteristics of a Parent Function
Answer: D and R: no intercepts. The graph is symmetric about the origin, so f (x) is odd. The graph is continuous for all values in its domain with an infinite discontinuity at x = 0.The end behavior is as The graph decreases on both intervals of its domain.
Example 1

22. Describe the following characteristics of the graph of the parent function f (x) = x 2: domain, range, intercepts, symmetry, continuity, end behavior, and intervals on which the graph is increasing/decreasing.
Example 1

23. A. D: , R: ; y-intercept = (0, 0)
A. D: , R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , and as , The graph is decreasing on the interval and increasing on the interval
B. D: , R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , As , The graph is decreasing on the interval and increasing on the interval
C. D: , R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , As , The graph is decreasing on the interval and increasing on the interval
D. D: , R: ; no intercepts. The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , As , The graph is decreasing on the interval and increasing on the interval
Example 1

24. A. D: , R: ; y-intercept = (0, 0)
A. D: , R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , and as , The graph is decreasing on the interval and increasing on the interval
B. D: , R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , As , The graph is decreasing on the interval and increasing on the interval
C. D: , R: ; y-intercept = (0, 0). The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , As , The graph is decreasing on the interval and increasing on the interval
D. D: , R: ; no intercepts. The graph is symmetric with respect to the y-axis. The graph is continuous everywhere. The end behavior is as , As , The graph is decreasing on the interval and increasing on the interval
Example 1

25. Key Concept 5

26. A. Use the graph of f (x) = x 3 to graph the function g (x) = x 3 – 2.
Graph Translations
A. Use the graph of f (x) = x 3 to graph the function g (x) = x 3 – 2.
This function is of the form g (x) = f (x) – 2. So, the graph of g (x) is the graph of f (x) = x 3 translated 2 units down, as shown below.
Answer:
Example 2

27. A. Use the graph of f (x) = x 3 to graph the function g (x) = x 3 – 2.
Graph Translations
A. Use the graph of f (x) = x 3 to graph the function g (x) = x 3 – 2.
This function is of the form g (x) = f (x) – 2. So, the graph of g (x) is the graph of f (x) = x 3 translated 2 units down, as shown below.
Answer:
Example 2

28. Graph Translations
B. Use the graph of f (x) = x 3 to graph the function g (x) = (x – 1)3.
This function is of the form g (x) = f (x – 1) . So, g (x) is the graph of f (x) = x 3 translated 1 unit right, as shown below.
Answer:
Example 2

29. Graph Translations
B. Use the graph of f (x) = x 3 to graph the function g (x) = (x – 1)3.
This function is of the form g (x) = f (x – 1) . So, g (x) is the graph of f (x) = x 3 translated 1 unit right, as shown below.
Answer:
Example 2

30. Graph Translations
C. Use the graph of f (x) = x 3 to graph the function g (x) = (x – 1)3 – 2.
This function is of the form g (x) = f (x – 1) – 2. So, g (x) is the graph of f (x) = x 3 translated 2 units down and 1 unit right, as shown below.
Answer:
Example 2

31. Graph Translations
C. Use the graph of f (x) = x 3 to graph the function g (x) = (x – 1)3 – 2.
This function is of the form g (x) = f (x – 1) – 2. So, g (x) is the graph of f (x) = x 3 translated 2 units down and 1 unit right, as shown below.
Answer:
Example 2

32. Use the graph of f (x) = x 2 to graph the function g (x) = (x – 2)2 – 1.B. C. D.
Example 2

33. Use the graph of f (x) = x 2 to graph the function g (x) = (x – 2)2 – 1.B. C. D.
Example 2

34. Key Concept 6

35. The graph of g (x) is the graph of translated 1 unit up. So, .
Write Equations for Transformations
A. Describe how the graphs of and g (x) are related. Then write an equation for g (x).
The graph of g (x) is the graph of translated 1 unit up. So,
Answer:
Example 3

36. The graph of g (x) is the graph of translated 1 unit up. So, .
Write Equations for Transformations
A. Describe how the graphs of and g (x) are related. Then write an equation for g (x).
The graph of g (x) is the graph of translated 1 unit up. So,
Answer: The graph is translated 1 unit up;
Example 3

37. Write Equations for Transformations
B. Describe how the graphs of and g (x) are related. Then write an equation for g (x).
The graph of g (x) is the graph of translated 1 unit to the left and reflected in the x-axis. So,
Example 3

38. Write Equations for Transformations
Answer:
Example 3

39. Write Equations for Transformations
Answer: The graph is translated 1 unit to the left and reflected in the x-axis;
Example 3

40. A. The graph is translated 3 units up; g (x) = x 3 + 3.
Describe how the graphs of f (x) = x 3 and g (x) are related. Then write an equation for g (x).
A. The graph is translated 3 units up; g (x) = x
B. The graph is translated 3 units down; g (x) = x 3 – 3.
C. The graph is reflected in the x-axis; g (x) = –x 3.
D. The graph is translated 3 units down and reflected in the x-axis; g (x) = –x 3 – 3.
Example 3

41. A. The graph is translated 3 units up; g (x) = x 3 + 3.
Describe how the graphs of f (x) = x 3 and g (x) are related. Then write an equation for g (x).
A. The graph is translated 3 units up; g (x) = x
B. The graph is translated 3 units down; g (x) = x 3 – 3.
C. The graph is reflected in the x-axis; g (x) = –x 3.
D. The graph is translated 3 units down and reflected in the x-axis; g (x) = –x 3 – 3.
Example 3

42. Key Concept 7

43. Describe and Graph Transformations
A. Identify the parent function f (x) of , and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes.
The graph of g(x) is the same as the graph of the reciprocal function expanded vertically because and 1 < 3.
Example 4

44. Describe and Graph Transformations
Answer:
Example 4

45. Describe and Graph Transformations
Answer: ; g (x) is represented by the expansion of f (x) vertically by a factor of 3.
Example 4

46. Describe and Graph Transformations
B. Identify the parent function f (x) of g (x) = –|4x|, and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes.
The graph of g (x) is the same as the graph of the absolute value function f (x) = |x| compressed horizontally and then reflected in the x-axis because g (x) = –4(|x|) = –|4x| = –f (4x), and 1 < 4.
Example 4

47. Describe and Graph Transformations
Answer:
Example 4

48. Describe and Graph Transformations
Answer: f (x) = |x| ; g (x) is represented by the compression of f (x) horizontally by a factor of 4 and reflection in the x-axis.
Example 4

49. Identify the parent function f (x) of g (x) = – (0
Identify the parent function f (x) of g (x) = – (0.5x)3, and describe how the graphs of g (x) and f (x) are related. Then graph f (x) and g (x) on the same axes.
Example 4

50. A. f (x) = x 3; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of .
B. f (x) = x 3; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of and reflected in the x-axis.
C. f (x) = x 3; g(x) is represented by the reflection of the graph of f (x) in the x-axis.
D. f (x) = x 2; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of and reflected in the x-axis.
Example 4

51. A. f (x) = x 3; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of .
B. f (x) = x 3; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of and reflected in the x-axis.
C. f (x) = x 3; g(x) is represented by the reflection of the graph of f (x) in the x-axis.
D. f (x) = x 2; g(x) is represented by the expansion of the graph of f (x) horizontally by a factor of and reflected in the x-axis.
Example 4

52. On the interval , graph y = |x + 2|.
Graph a Piecewise-Defined Function
Graph
On the interval , graph y = |x + 2|.
On the interval [0, 2], graph y = |x| – 2.
On the interval graph
Draw circles at (0, 2) and (2, 2); draw dots at (0, –2) and (2, 0) because f (0) = –2, and f (2) = 0, respectively.
Example 5

53. Graph a Piecewise-Defined Function
Answer:
Example 5

54. Graph a Piecewise-Defined Function
Answer:
Example 5

55. Graph the function A. B. C. D.
Example 5

56. Graph the function A. B. C. D.
Example 5

57. Transformations of Functions
A. AMUSEMENT PARK The “Wild Ride” roller coaster has a section that is shaped like the function , where g (x) is the vertical distance in yards the roller coaster track is from the ground and x is the horizontal distance in yards from the start of the ride. Describe the transformations of the parent function f (x) = x 2 used to graph g (x).
Example 6

58. Transformations of Functions
Rewrite the function so that it is in the form g(x) = a(x – h)2 + k by completing the square.
Original function
Factor
Complete the square.
Example 6

59. Write x 2 – 100x + 2500 as a perfect square and simplify.
Transformations of Functions
Write x 2 – 100x as a perfect square and simplify.
Answer:
Example 6

60. Write x 2 – 100x + 2500 as a perfect square and simplify.
Transformations of Functions
Write x 2 – 100x as a perfect square and simplify.
Answer: g (x) is the graph of f (x) translated 50 units right, compressed vertically, reflected in the x-axis, and then translated 50 units up
Example 6

61. Transformations of Functions
B. AMUSEMENT PARK The “Wild Ride” roller coaster has a section that is shaped like the function , where g (x) is the vertical distance in yards the roller coaster track is from the ground and x is the horizontal distance in yards from the start of the ride. Suppose the ride designers decide to increase the highest point of the ride to 70 yards. Rewrite g (x) to reflect this change. Graph both functions on the same coordinate axes using a graphing calculator.
Example 6

62. Transformations of Functions
A change of the height of the highest point of the ride from 50 yards to 70 yards is a vertical translation, so add an additional 20 yards to the expression in part A to get The graph of both functions on the same coordinate axes using a graphing calculator is shown.
Example 6

63. Transformations of Functions
Answer:
Example 6

64. Transformations of Functions
Answer:
Example 6

65. STUNT RIDING A stunt motorcyclist jumps from ramp to ramp according to the model shaped like the function , where g (x) is the vertical distance in feet the motorcycle is from the ground and x is the horizontal distance in feet from the start of the jump. Describe the transformations of f (x) = x 2 used to graph g (x).
Example 6

66. A. The graph of g (x) is the graph of f (x) translated 75 units right.
B. The graph of g (x) is the graph of f (x) translated 18 units up, compressed vertically, and reflected in the x-axis.
C. The graph of g (x) is the graph of f (x) translated 75 units right and reflected in the x-axis.
D. The graph of g (x) is the graph of f (x) translated 75 units right and 18 units up, compressed vertically, and reflected in the x-axis.
Example 6

67. A. The graph of g (x) is the graph of f (x) translated 75 units right.
B. The graph of g (x) is the graph of f (x) translated 18 units up, compressed vertically, and reflected in the x-axis.
C. The graph of g (x) is the graph of f (x) translated 75 units right and reflected in the x-axis.
D. The graph of g (x) is the graph of f (x) translated 75 units right and 18 units up, compressed vertically, and reflected in the x-axis.
Example 6

68. Key Concept 8

69. Describe and Graph Transformations
A. Use the graph of f (x) = x 2 – 4x + 3 to graph the function g(x) = |f (x)|.
Example 7

70. Describe and Graph Transformations
The graph of f (x) is below the x-axis on the interval (1, 3), so reflect that portion of the graph in the x-axis and leave the rest unchanged.
Answer:
Example 7

71. Describe and Graph Transformations
The graph of f (x) is below the x-axis on the interval (1, 3), so reflect that portion of the graph in the x-axis and leave the rest unchanged.
Answer:
Example 7

72. Describe and Graph Transformations
B. Use the graph of f (x) = x 2 – 4x + 3 to graph the function h (x) = f (|x|).
Example 7

73. Describe and Graph Transformations
Replace the graph of f (x) to the left of the y-axis with a reflection of the graph to the right of the y-axis.
Answer:
Example 7

74. Describe and Graph Transformations
Replace the graph of f (x) to the left of the y-axis with a reflection of the graph to the right of the y-axis.
Answer:
Example 7

75. Use the graph of f (x) shown to graph g(x) = |f (x)| and h (x) = f (|x|).
Example 7

76. A. B. C. D.
Example 7

77. A. B. C. D.
Example 7

78. End of the Lesson