The Algebra Teacher\'s Guide to Reteaching Essential Concepts and Skills

Teaching Notes 8.8: Describing Dilations of the Graph of a

Function

Unlike a reflection and vertical or horizontal shift whose graphs have the same size and shape, a

dilation either stretches or compresses a graph. Students often have trouble recognizing the value

that indicates the stretching or compressing of a graph.

1. Explain that a dilation can stretch a graph away from thex-axis or compress (or shrink) a
   graph toward thex-axis.


2. Show the graphs of the following basic functions to your students: identity,f(x)=x;squar-
   ing,f(x)=x^2 ;cubing,f(x)=x^3 ; square root,f(x)=

√

x; absolute value,f(x)=|x|;and

reciprocal,f(x)=

1

x

.

3. Sketch the graphs of the following functionsby drawing t-tables and plotting points:
   f(x)=x^2 andg(x)= 3 x^2. Explain that they-coordinate of each ordered pair ing(x)is3times
   the value of the correspondingy-coordinate off(x). The graph ofgis a stretching of the
   graph off.


4. Sketch the graphs of the following functionsby drawing t-tables and plotting points:
   f(x)=x^2 andh(x)=

1

3

x^2 or

x^2

3

. Explain that they-coordinate of each ordered pair inh(x)is
1
3
the value of the correspondingy-coordinate off(x). The graph ofhis a compression of the
graph off.
5. Review the information and example on the worksheet with your students. Emphasize that
students must identify the basic function and whether it is a stretch or compression by
looking at the value ofa. Note that the domains of the functions are restricted so that the
functions are defined.

EXTRA HELP:

Always rewrite the function with a numerical coefficient.

ANSWER KEY:

The graph shows the following:

(1)Reciprocal function stretched (2)Absolute value function compressed

(3)Cubing function stretched (4)Identity function compressed

(5)Squaring function stretched (6)Square root function compressed

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(Challenge)Louis is correct. A negative number represents a reflection in thex-axis and 2

shows a stretch of the graph.

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288 THE ALGEBRA TEACHER’S GUIDE