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Transformations of Functions. 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. SECTION 2.7. 1. 2. 3. 4. TRANSFORMATIONS. - PowerPoint PPT Presentation
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Transformations of FunctionsSECTION 2.7
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2
3
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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 .
1
2y
1
2
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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.