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