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Transformations of FunctionsSECTION 2.7

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Learn the meaning of transformations.

Use vertical or horizontal shifts to graph functions.

Use reflections to graph functions.

Use stretching or compressing to graph functions.

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TRANSFORMATIONS

If a new function is formed by performing certain operations on a given function f , then the graph of the new function is called a transformation of the graph of f.

Parent Functions – The simplest function of its kind. All other functions of its kind are Transformations of the parent.

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EXAMPLE 1 Graphing Vertical Shifts

Let , 2, and 3.f x x g x x h x x Sketch the graphs of these functions on the same coordinate plane. Describe how the graphs of g and h relate to the graph of f.

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EXAMPLE 1 Graphing Vertical Shifts

SolutionMake a table of values.

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EXAMPLE 1 Graphing Vertical Shifts

Solution continued

Graph the equations.The graph of y = |x| + 2 is the graph of y = |x| shifted two units up. The graph of y = |x| – 3 is the graph of y = |x|shifted three units down.

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VERTICAL SHIFT

Let d > 0. The graph of y = f (x) + d is the graph of y = f (x) shifted d units up, and the graph of y = f (x) – d is the graph of y = f (x) shifted d units down.

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EXAMPLE 2 Writing Functions for Horizontal Shifts

Let f (x) = x2, g(x) = (x – 2)2, and h(x) = (x + 3)2.

A table of values for f, g, and h is given on the next slide. The graphs of the three functions f, g, and h are shown on the following slide.

Describe how the graphs of g and h relate to the graph of f.

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EXAMPLE 2 Writing Functions for Horizontal Shifts

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EXAMPLE 2 Writing Functions for Horizontal Shifts

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EXAMPLE 2 Writing Functions for Horizontal Shifts

All three functions are squaring functions.Solution

The x-intercept of f is 0.The x-intercept of g is 2.

a. g is obtained by replacing x with x – 2 in f .

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f x

g x

x

x

For each point (x, y) on the graph of f , there will be a corresponding point (x + 2, y) on the graph of g. The graph of g is the graph of f shifted 2 units to the right.

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EXAMPLE 2 Writing Functions for Horizontal Shifts

Solution continued

The x-intercept of f is 0.The x-intercept of h is –3.

b. h is obtained by replacing x with x + 3 in f .

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f x

h x

x

x

For each point (x, y) on the graph of f , there will be a corresponding point (x – 3, y) on the graph of h. The graph of h is the graph of f shifted 3 units to the left.

The tables confirm both these considerations.

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HORIZONTAL SHIFT

The graph of y = f (x – c) is the graph of y = f (x) shifted |c| units to the right, if c > 0, to the left if c < 0.

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EXAMPLE 3

Sketch the graph of the function

2 3.g x x

Solution

Identify and graph the parent function

.f x x

Graphing Combined Vertical and Horizontal Shifts

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EXAMPLE 3

Solution continued

Graphing Combined Vertical and Horizontal Shifts

2 3.g x x

Translate 2 units to the left

Translate 3 units down

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REFLECTION IN THE x-AXISThe graph of y = – f (x) is a reflection of the graph of y = f (x) in the x-axis.

If a point (x, y) is on the graph of y = f (x), then the point (x, –y) is on the graph of y = – f (x).

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REFLECTION IN THE x-AXIS

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REFLECTION IN THE y-AXIS

The graph of y = f (–x) is a reflection of the graph of y = f (x) in the y-axis.

If a point (x, y) is on the graph of y = f (x), then the point (–x, y) is on the graph of y = f (–x).

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REFLECTION IN THE y-AXIS

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EXAMPLE 4 Combining Transformations

Explain how the graph of y = –|x – 2| + 3 can be obtained from the graph of y = |x|.

Solution

Step 1 Shift the graph of y = |x| two units right to obtain the graph of y = |x – 2|.

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EXAMPLE 4 Combining Transformations

Solution continued

Step 2 Reflect the graph of y = |x – 2| in the x–axis to obtain the graph of y = –|x – 2|.

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EXAMPLE 4 Combining Transformations

Solution continued

Step 3 Shift the graph of y = –|x – 2| three units up to obtain the graph of y = –|x – 2| + 3.

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EXAMPLE 5Stretching or Compressing a Function Vertically

Solution

Sketch the graphs of f, g, and h on the same coordinate plane, and describe how the graphs of g and h are related to the graph of f.

f x x , g x 2 x , and h x 1

2x .Let

x –2 –1 0 1 2

f(x) 2 1 0 1 2

g(x) 4 2 0 2 4

h(x) 1 1/2 0 1/2 1

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EXAMPLE 5Stretching or Compressing a Function Vertically

Solution continued

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EXAMPLE 5Stretching or Compressing a Function Vertically

Solution continued

The graph of y = 2|x| is the graph of y = |x| vertically stretched (expanded) by multiplying each of its y–coordinates by 2.

The graph of |x| is the graph of y = |x|

vertically compressed (shrunk) by multiplying

each of its y–coordinates by .

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2y

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VERTICAL STRETCHING OR COMPRESSING

The graph of y = a f (x) is obtained from the graph of y = f (x) by multiplying the y-coordinate of each point on the graph of y = f (x) by a and leaving the x-coordinate unchanged. The result is

1. A vertical stretch away from the x-axis if a > 1;

2. A vertical compression toward the x-axis if 0 < a < 1.

If a < 0, the graph of f is first reflected in the x-axis, then vertically stretched or compressed.