sec 2.4 transformation of graphs. copyright © by houghton mifflin company, inc. all rights...

Sec 2.4Transformation of Graphs

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 1: 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


Ex 2. Match the functions with the correct graph. 2,y x 2 1, andy x 2 3.y x

–5 5

4

–4


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

Ex 3: 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


Ex 4: Describe the changes in the graph of the function to

First make a vertical shift 4 units downward.

Then a horizontal shift 5 units left.

45 xyxy


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

Ex 5: 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)


HW #4 p190 1-4, 7, 11-13, 16, 23, 24, 27


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.

Ex 1: 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


- 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.

Ex 2: y = |2x| is the graph of y = |x| shrunk horizontally by 2.

xy21

is the graph of y = |x| stretched horizontally by .

xy21

21


Ex 3: Match the function to its graph. 3,y x 3, d0 an1 y x 3.1

10y x

–5 5

5

–5


Ex 4: Explain how the graph of is obtained

from y = x3

3)1(21 3 xy


HW #5 p190 5-10, 15, 17, 18, 25,

26, 28, 30, 32

sec 2.4 transformation of graphs. copyright © by houghton mifflin company, inc. all rights...

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