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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

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

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

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

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

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

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

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

Key Concept 1

Key Concept 2

Key Concept 3

Key Concept 4

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

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

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

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

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

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

Key Concept 5

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

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

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

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

Key Concept 6

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

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

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

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 3 + 3. 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

Key Concept 7

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

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

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

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

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

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

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

Graph a Piecewise-Defined Function Answer: Example 5

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

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

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

Write x 2 – 100x + 2500 as a perfect square and simplify. Transformations of Functions Write x 2 – 100x + 2500 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

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

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

Transformations of Functions Answer: Example 6

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

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

Key Concept 8

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

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

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

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

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

A. B. C. D. Example 7

End of the Lesson