SECTION 2.7
Transformations of Functions
1
Learn the meaning of transformations.
2
Use vertical or horizontal shifts to graph
functions.
3
Use reflections to graph functions.
4
Use stretching or compressing to graph
functions.
1
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.
2
Parent Functions – The simplest function of its kind. All
other functions of its kind are Transformations of the
parent.
EXAMPLE 1
Graphing Vertical Shifts
Let f  x   x , g  x   x  2, and h  x   x  3.
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.
8
EXAMPLE 1
Solution
Graphing Vertical Shifts
Make 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.
10
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
13
EXAMPLE 2
Writing Functions for Horizontal Shifts
14
EXAMPLE 2
Writing Functions for Horizontal Shifts
Solution
All three functions are squaring functions.
a. g is obtained by replacing x with x – 2 in f .
f  x 
x2
g  x    x  2
2
The x-intercept of f is 0.
The x-intercept of g is 2.
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
b. h is obtained by replacing x with x + 3 in f .
f  x 
x2
h  x    x  3
2
The x-intercept of f is 0.
The x-intercept of h is –3.
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.
16
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
Graphing Combined Vertical and
Horizontal Shifts
Sketch the graph of the function
g  x   x  2  3.
Solution
Identify and graph the parent function
f  x   x.
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EXAMPLE 3
Graphing Combined Vertical and
Horizontal Shifts
Solution continued
g  x   x  2  3.
Translate 2 units to the left
Translate 3 units down
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REFLECTION IN THE x-AXIS
The 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).
21
REFLECTION IN THE x-AXIS
22
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).
23
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|.
25
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|.
26
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 5
Stretching or Compressing a Function
Vertically
1
Let f x   x , g x   2 x , and h x   x .
2
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.
Solution
x
f(x)
g(x)
h(x)
–2
2
4
1
–1
1
2
1/2
0
0
0
0
1
1
2
1/2
2
2
4
1
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EXAMPLE 5
Stretching or Compressing a Function
Vertically
Solution continued
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EXAMPLE 5
Stretching 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.
1
The graph ofy 
|x| is the graph of y = |x|
2
vertically compressed (shrunk) by multiplying
1
each of its y–coordinates by .
2
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VERTICAL STRETCHING OR
COMPRESSING
The graph of y = af(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.
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