Transcript 2.9
2-9
Absolute–Value Functions LEARNING GOALS FOR LESSON 2.9
Write, graph and transform absolute-value functions including (1) translations, (2) reflections, and (3) stretches/compressions.
The graph of the parent absolute-value function
f
(
x
) = |
x
| has a
V
shape with a minimum point or vertex at (0, 0).
Remember!
The general forms for translations are Vertical:
g
(
x
) =
f
(
x
) +
k
Horizontal: g(
x
) =
f
(
x
–
h
)
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Absolute–Value Functions
Example 1A: Translating Absolute-Value Functions
new equation and tell what the vertex is.
5 units down
Example 1B: Translating Absolute-Value Functions
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
LG 2.9.1
1 unit left
The vertex of g(x) is (___, ___)
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Absolute–Value Functions
Example 1C: Translations of an Absolute-Value Function Translate f(x) = |x| so that the vertex is at (–1, –3). Then graph.
LG 2.9.1
Check Yourself!
Translate f(x) = |x| so that the vertex is at (4, –2). Then graph.
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Absolute–Value Functions
Absolute-value functions can also be stretched, compressed, and reflected.
Remember!
Reflection across
x
-axis:
g
(
x
) = –
f
(x) Reflection across
y
-axis: g(x) =
f
( –
x
)
Perform the transformation. Then graph.
Reflect the graph. f(x) =|x – 2| + 3 across the y-axis.
g
LG 2.9.2
f
Remember!
Vertical stretch and compression :
g
(
x
) =
a f
(x) Horizontal stretch and compression: g(x) =
f
LG 2.9.3
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Absolute–Value Functions
Stretch the graph. f(x) = |x| – 1 vertically by a factor of 2.
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Absolute–Value Functions
Example 3C: Transforming Absolute-Value Functions
Compress the graph of f(x) = |x + 2| – 1 horizontally by a factor of .
LG 2.9.1
Perform the transformation. Then graph.
Check Yourself!
Reflect the graph. f(x) = –|x – 4| + 3 across the y-axis.
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Absolute–Value Functions
Lesson Quiz: Part I Perform each transformation. Then graph.
1.
Translate
f
(
x
) = |
x
| 3 units right.
2.
Translate
f
(
x
) = |
x
| so the vertex is at (2, –1).
3.
Stretch the graph of
f
(
x
) = |2
x
| – 1 vertically by a factor of 3 and reflect it across the
x
-axis.