Shifting Graphs - Math

Digital Lesson

Shifting Graphs

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.

y y = x2 + 3

8

y = x2

4

-4 -4

-8

x

4

y = -x2

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2

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

f(x)

-c

f(x) ? c

x

Copyright ? by Houghton Mifflin Company, Inc. All rights reserved.

3

Example: Use the graph of f (x) = |x| to graph the functions f (x) = |x| + 3 and f (x) = |x| ? 4.

y

8 4

-4 -4

f (x) = |x| + 3 f (x) = |x|

f (x) = |x| ? 4

x

4

Copyright ? by Houghton Mifflin Company, Inc. All rights reserved.

4

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

-c +c

number, then the

graph of f (x + c) is

the graph of y = f (x)

shifted to the left c units.

x y = f (x + c) y = f (x) y = f (x ? c)

Copyright ? by Houghton Mifflin Company, Inc. All rights reserved.

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