Page 738 - Algebra 1

Reflections, Stretches, and Compressions of Absolute-Value Graphs
The absolute-value parent function can be reflected, stretched, and compressed by multiplying by a constant a.
If a < 0, then the graph is reflected across the x-axis.
If ⎢a > 1, then the graph is stretched vertically, or away from the x-axis. If ⎢a < 1, then the graph is compressed vertically, or toward the x-axis.
Math Reasoning
Formulate What values are described by the inequality
⎢a > 1? ⎢a < 1?
Example
4
Reflecting, Stretching, and Compressing Absolute-Value Graphs
Describe the graph of each function.
a. f(x) = 3⎢x
SOLUTION
a = 3, so ⎢a = 3.
Since ⎢a > 1, the graph is stretched vertically.
b. f(x) = -4⎢x
SOLUTION
a = -4, so ⎢a = 4.
Since a < 0, the graph is reflected across the x-axis. Since ⎢a > 1, the graph is stretched vertically.
c. f(x) = -0.2⎢x
SOLUTION
a = -0.2, so ⎢a = 0.2.
Since a < 0, the graph is reflected across the x-axis. Since ⎢a < 1, the graph is compressed vertically.
Application: Travel
A helicopter pilot is flying from town A to town B at 60 miles per hour. To make sure he is on course, he will fly over a landmark that he knows is 20 miles from town A. Write and graph the distance from the landmark as a function of minutes of flight time.
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-8
-4
-4
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-8
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-8
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Hint
A function that is reflected can also be stretched or compressed.
y
8
Math Reasoning
Connect What other function is stretched or compressed vertically by changing the a value?
4
O
x
-4
-8
Example
5
SOLUTION
y
Let a = rate = 60 mph = 1 mile per minute
Let h = time from landmark = _20 = 20 minutes
1
Let k = closest distance to landmark = 0 miles
-80 -40
80
O
x
f(x) = 1⎢x - 20 + 0 f(x) = 1⎢x - 20
-40
y
Lesson 107 723
4
0
8
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-80