Shifting Graphs Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved....

Shifting Graphs

Digital Lesson

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

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


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


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


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


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.


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


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


-4

y

4

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)


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


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)


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

4

1xy

4

1

2

4

1xy

y = 2x2


- 4x

y

4

4

y = |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.

xy2

1

is the

graph of y = |x| stretched

horizontally by .

xy2

1

2

1


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

3110

y x


- 4

4

4

8

x

y

Example: Graph using the graph of y = x3.3)1(2

1 3 xy

3)1(2

1 3 xyStep 4:

- 4

4

4

8

x

y

Step 1: y = x3

Step 2: y = (x + 1)3

3)1(2

1 xyStep 3:

Graph y = x3 and do one transformation at a time.

Shifting Graphs Digital Lesson. Copyright © by Houghton Mifflin Company, Inc. All rights reserved....

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