Copyright © 2011 Pearson Education, Inc. Slide 2.2-1

2.2 Vertical Translations of Graphs

Vertical Shifting of the Graph of a Function

If the graph of is obtained by shifting the graph of upward a distance of c units. The graph of is obtained by shifting the graph of downward a distance of c units.

,0c cxfy )()(xfy cxfy )(

)(xfy

Copyright © 2011 Pearson Education, Inc. Slide 2.2-2

2.2 Example of a Vertical Shift with the Calculator

Give the equation of each function graphed.

Solution:a. b.

units. 5down shifted is ofgraph thesince 5 xyxy units. 3 up shifted is ofgraph thesince 3 xyxy

Figure 21b pg 2-30

Copyright © 2011 Pearson Education, Inc. Slide 2.2-3

2.2 Horizontal Translations of Graphs

Horizontal Shifting of the Graph of a Function

If the graph of is obtained by shifting the graph of to the right a distance of c units. The graph of is obtained by shifting the graph of to the left a distance of c units.

,0c )( cxfy )(xfy )( cxfy

)(xfy

Figure 22 pg 2-31

Copyright © 2011 Pearson Education, Inc. Slide 2.2-4

2.2 Example of a Horizontal Shift

Give the equation of each function in the graphs below.

Solution:

a.

b.

4 since the graph of is shifted right 4 units.y x y x 2 since the graph of is shifted left 2 units.y x y x

Figure 23b pg 2-34

Copyright © 2011 Pearson Education, Inc. Slide 2.2-5

2.2 Example of Vertical and Horizontal Shifts

• Describe how the graph of would be obtained by translating the graph of Sketch the graphs on the same xy-plane.

Solution is shifted 2 units left and 6 units down.

The graph changes from decreasing to increasing at the point (2, 6).

2 6y x . xy

2 6y x

y = |x+ 2| 6

y = |x|

Copyright © 2011 Pearson Education, Inc. Slide 2.2-6

2.2 Effects of Shifts on Domain and Range

• The domains and ranges of functions may or may not be affected by translations.– If the domain (or range) is a horizontal (or vertical) shift

will not affect the domain (or range).

– If the domain (or range) is not a horizontal (or vertical) shift will affect the domain (or range).

• Example Determine the domain and range of the following shifted graph.

),,(

),,(

).[-30, is range its and )[20, isfunction

theofdomain theSo units. 30down and

units 20right graph basic theshiftingby

obtained is 3020 ofgraph The

).[0,domain has graph basic The

xy

xy

Copyright © 2011 Pearson Education, Inc. Slide 2.2-7

2.2 Applying a Horizontal Shift to an Equation Model

The table lists U.S. sales of Toyota vehicles in millions.

a. Find the corresponding equation that allows direct input of the year.

b. Use the equation to estimate U.S. sales of Toyota vehicles in 2007.

Solution

Year Vehicles

2000 1.6

2001 1.7

2002 1.8

2003 1.9

2004 2.0

The data can be modeled by the equation y = 0.1x + 1.6, where x = 0 corresponds to 2000, x = 1 to 2001, and so on.

a. Because 2000 corresponds to 0, the graph of y = 0.1x + 1.6 would have to be shifted 2000 units to the right. Thus, the equation becomes y = 0.1(x – 2000) + 1.6.

b. y = 0.1(2007 – 2000) + 1.6 = 2.3 The model estimates U.S. sales of Toyota vehicles in 2007 to be about 2.3 million.