Shifting Graphs
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As you saw with the Nspires, the graphs of many functions are transformations of the graphs of very basic functions.
The graph of y = –x2 is the reflection of the graph of y = x2 in the x-axis.
Example: The graph of y = x2 + 3 is the graph of y = x2 shifted upward three units. This is a vertical shift.
x
y
-4 4
4
-4
-8
8
y = –x2
y = x2 + 3
y = x2
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f (x)
f (x) + c
+c
f (x) – c-c
If c is a positive real number, the graph of f (x) + c is the graph of y = f (x) shifted upward c units.
Vertical Shifts
If c is a positive real number, the graph of f (x) – c is the graph of y = f(x) shifted downward c units.
x
y
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h(x) = |x| – 4
Example: Use the graph of f (x) = |x| to graph the functions g(x) = |x| + 3 and h(x) = |x| – 4.
f (x) = |x|
x
y
-4 4
4
-4
8 g(x) = |x| + 3
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Graphing Utility: Sketch the graphs given by 2,y x 2 1, andy x 2 3.y x
–5 5
4
–4
2+1 y x
2y x
2 3y x
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x
y
y = f (x) y = f (x – c)
+c
y = f (x + c)
-c
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.
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 left c units.
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f (x) = x3
h(x) = (x + 4)3
Example: Use the graph of f (x) = x3 to graph g (x) = (x – 2)3 and h(x) = (x + 4)3 .
x
y
-4 4
4
g(x) = (x – 2)3
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Graphing Utility: Sketch the graphs given by 2,y x 2( 3) , andy x 2( 1) .y x
–5 6
7
–1
2( 3)y x
2y x
2( 1)y x
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-4
y4
x-4
x
y4
Example: Graph the function using the graph of .
First make a vertical shift 4 units downward.
Then a horizontal shift 5 units left.
45 xyxy
(0, 0)(4, 2)
(0, – 4)(4, –2)
xy
4 xy
45 xy
(– 5, –4) (–1, –2)
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y = f (–x) y = f (x)
y = –f (x)
The graph of a function may be a reflection of the graph of a basic function.
The graph of the function y = f (–x) is the graph of y = f (x) reflected in the y-axis.
The graph of the function y = –f (x) is the graph of y = f (x) reflected in the x-axis.
x
y
Copyright © by Houghton Mifflin Company, Inc. All rights reserved.
x
y
4
4y = x2
y = – (x + 3)2
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.
x
y
– 4 4
4
-4
y = – x2
(–3, 0)
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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.
Example: y = 2x2 is the graph of y = x2 stretched vertically by 2.
– 4x
y
4
4
y = x2
is the graph of y = x2
shrunk vertically by .
2
41 xy
41
2
41 xy
y = 2x2
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- 4x
y
4
4y = |x|
y = |2x|
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.
Example: y = |2x| is the graph of y = |x| shrunk horizontally by 2.
xy21
is the graph of y = |x| stretched horizontally by .
xy21
21
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Graphing Utility: Sketch the graphs given by 3,y x 3, d0 an1 y x 3.1
10y x
–5 5
5
–5
310y x
3y x
3110y x
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- 4
4
4
8
x
y
Example: Graph using the graph of y = x3.3)1(21 3 xy
3)1(21 3 xyStep 4:
- 4
4
4
8
x
y
Step 1: y = x3
Step 2: y = (x + 1)3
3)1(21
xyStep 3:
Graph y = x3 and do one transformation at a time.
Graphing Functions (Cont.)
( ) 9f x x
Without a calculator, draw a quick sketch of each function.
( ) | 3 | 9f x x ( ) | 4 | 8f x x ( ) | | 2f x x 2( ) 2 ( 5)f x x 3( ) 7f x x 2( ) 12f x x ( ) 4 8f x x