1 transformations of functions section 2.7 1 2 3 4 learn the meaning of transformations. use...
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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
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
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
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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 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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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
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.