Shifting Graphs
As you saw with the Nspires, the graphs of many functions
are transformations of the graphs of very basic functions.
Example:
The graph of y = x2 + 3
is the graph of y = x2
shifted upward three units.
This is a vertical shift.
The graph of y = –x2 is the
reflection of the graph of
y = x2 in the x-axis.
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y
y = x2 + 3
8
y = x2
4
x
4
-4
-4
y = –x2
-8
Vertical Shifts
If c is a positive real number, the graph of f (x) + c
is the graph of y = f (x) shifted upward c units.
If c is a positive real number, the graph of f (x) – c
is the graph of y = f(x) shifted downward c units.
y
f (x) + c
+c
-c
f (x)
f (x) – c
x
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Example: Use the graph of f (x) = |x| to graph the
functions g(x) = |x| + 3 and h(x) = |x| – 4.
y
g(x) = |x| + 3
8
f (x) = |x|
4
h(x) = |x| – 4
x
4
-4
-4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Graphing Utility: Sketch the graphs given by
y  x 2, y  x 2  1, and y  x 2  3.
4
y  x 2+1
–5
5
y  x2
y  x2  3
–4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Horizontal Shifts
If c is a positive real number, then the graph of
f (x – c) is the graph of y = f (x) shifted to the right
c units.
y
If c is a positive real
number, then the
graph of f (x + c) is
the graph of y = f (x)
shifted to the left
c units.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
-c
+c
y = f (x + c)
y = f (x)
x
y = f (x – c)
Example: Use the graph of f (x) = x3 to graph
g (x) = (x – 2)3 and h(x) = (x + 4)3 .
y
f (x) = x3
4
-4
h(x) = (x + 4)3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
x
g(x) = (x – 2)3
Graphing Utility: Sketch the graphs given by
y  x 2, y  (x  3)2, and y  (x  1)2.
7
y  (x  3)2
y  (x  1)2
y  x2
–5
6
–1
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph the function y  x  5  4
using the graph of y  x .
First make a vertical shift
4 units downward.
Then a horizontal shift 5
units left.
y
y
4
(0, 0)
(4, 2)
y x
4
y  x5 4
x
x
-4
-4
(0, – 4)
(4, –2)
y  x 4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
(– 5, –4)
(–1, –2)
The graph of a function may be a reflection of the
graph of a basic function.
y
The graph of the
function y = f (–x)
is the graph of
y = f (x) reflected
in the y-axis.
y = f (–x)
y = f (x)
x
y = –f (x)
The graph of the function y = –f (x)
is the graph of y = f (x) reflected in the x-axis.
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph y = –(x + 3)2 using the graph of y = x2.
First reflect the graph
in the x-axis.
Then shift the graph
three units to the left.
y
4
y
y = x2
4
(–3, 0)
x
–4
4
-4
y = –x2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
x
4
y = –(x + 3)2
Vertical Stretching and Shrinking
If c > 1 then the graph of y = c f (x) is the graph of y = f (x)
stretched vertically by c.
If 0 < c < 1 then the graph of y = c f (x) is the graph of y = f (x)
shrunk vertically by c.
y = x2
Example: y = 2x2 is the
graph of y = x2
stretched vertically by 2.
1 2
y  x is the graph of y = x2
4
shrunk vertically by 1 .
4
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y
y = 2x2
4
y
–4
1 2
x
4
x
4
Horizontal Stretching and Shrinking
If c > 1, the graph of y = f (cx) is the graph of y = f (x)
shrunk horizontally by c.
If 0 < c < 1, the graph of y = f (cx) is the graph of y = f (x)
stretched horizontally by c.
y
Example: y = |2x| is the
graph of y = |x| shrunk
horizontally by 2.
1
y  x is the
2
graph of y = |x| stretched
1
horizontally by .
2
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
y = |2x|
y = |x|
4
1
y x
2
x
-4
4
Graphing Utility: Sketch the graphs given by
3
1
y

x
.
y  x , y  10x , and
10
3
3
5
y  10x
3
y  1 x3
10
–5
5
y  x3
–5
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
Example: Graph y 
1
( x  1)3  3 using the graph of y = x3.
2
Graph y = x3 and do one transformation at a time.
y
y
8
8
4
4
x
x
4
-4
Step 1: y = x3
Step 2: y = (x + 1)3
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
4
-4
1
( x  1) 3
2
1
Step 4: y  ( x  1)3  3
2
Step 3: y 
Graphing Functions
Graphing Functions (Cont.)
Without a calculator, draw a quick
sketch of each function.
fff(f((fxx(xf()fx)x)(f()


|


|
|
x
x

|
3

4
2
|
|


9
8
x

)
2


12
(
x


x
5)

8
x( x))  x x 
 97
x 3 4
2 2