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Holt Algebra 2
2-9 Absolute–Value Functions
Graph and transform absolute-value functions.
Objective
An absolute-value function is a function whose rule contains an absolute-value expression. The graph of the parent absolute-value function f(x) = |x| has a V shape with a minimum point or vertex at (0, 0).
Holt Algebra 2
2-9 Absolute–Value Functions
The absolute-value parent function is composed of two linear pieces, one with a slope of –1 and one with a slope of 1. In Lesson 2-6, you transformed linear functions. You can also transform absolute-value functions.
Holt Algebra 2
2-9 Absolute–Value Functions
The general forms for translations are
Vertical:
g(x) = f(x) + k
Horizontal:
g(x) = f(x – h)
Remember!
Holt Algebra 2
2-9 Absolute–Value Functions
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
5 units down
The graph of g(x) = |x| – 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).
f(x) = |x|
g(x) = f(x) + k
g(x) = |x| – 5 f(x)
g(x)
Holt Algebra 2
2-9 Absolute–Value Functions
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
1 unit left
f(x) = |x|
g(x) = f(x – h )
g(x) = |x – (–1)| = |x + 1|
The graph of g(x) = |x + 1| is the graph of f(x) = |x| after a horizontal shift of 1 unit left. The vertex of g(x) is (–1, 0).
f(x)
g(x)
Holt Algebra 2
2-9 Absolute–Value Functions
4 units down
f(x) = |x|
g(x) = f(x) + k
g(x) = |x| – 4
Let g(x) be the indicated transformation of f(x) = |x|. Write the rule for g(x) and graph the function.
The graph of g(x) = |x| – 4 is the graph of f(x) = |x| after a vertical shift of 4 units down. The vertex of g(x) is (0, –4).
f(x)
g(x)
Holt Algebra 2
2-9 Absolute–Value Functions
Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).
2 units right
f(x) = |x|
g(x) = f(x – h)
g(x) = |x – 2| = |x – 2|
The graph of g(x) = |x – 2| is the graph of f(x) = |x| after a horizontal shift of 2 units right. The vertex of g(x) is (2, 0).
f(x)
g(x)
Holt Algebra 2
2-9 Absolute–Value Functions
Because the entire graph moves when shifted, the shift from f(x) = |x| determines the vertex of an absolute-value graph.
Holt Algebra 2
2-9 Absolute–Value Functions Translate f(x) = |x| so that the vertex is at
(–1, –3). Then graph.
g(x) = |x – h| + k
g(x) = |x – (–1)| + (–3)
g(x) = |x + 1| – 3
The graph of g(x) = |x + 1| – 3 is the graph of f(x) = |x| after a vertical shift down 3 units and a horizontal shift left 1 unit.
f(x)
g(x)
The graph confirms that the vertex is (–1, –3).
Holt Algebra 2
2-9 Absolute–Value Functions
Translate f(x) = |x| so that the vertex is at (4, –2). Then graph.
g(x) = |x – h| + k
g(x) = |x – 4| + (–2)
g(x) = |x – 4| – 2
The graph of g(x) = |x – 4| – 2 is the graph of f(x) = |x| after a vertical down shift 2 units and a horizontal shift right 4 units.
The graph confirms that the vertex is (4, –2).
g(x)
f(x)
Holt Algebra 2
2-9 Absolute–Value Functions
Reflection across x-axis: g(x) = –f(x)
Reflection across y-axis: g(x) = f(–x)
Remember!
Absolute-value functions can also be stretched, compressed, and reflected.
Vertical stretch and compression : g(x) = af(x)
Horizontal stretch and compression: g(x) = f
Remember!
Holt Algebra 2
2-9 Absolute–Value Functions
Perform the transformation. Then graph.
g(x) = f(–x)
g(x) = |(–x) – 2| + 3
Take the opposite of the input value.
Reflect the graph. f(x) =|x – 2| + 3 across the y-axis.
The vertex of the graph
g(x) = |–x – 2| + 3 is (–2, 3).
gf
Holt Algebra 2
2-9 Absolute–Value Functions
g(x) = af(x)
g(x) = 2(|x| – 1) Multiply the entire function by 2.
Stretch the graph. f(x) = |x| – 1 vertically by a factor of 2.
g(x) = 2|x| – 2
The graph of g(x) = 2|x| – 2 is the graph of f(x) = |x| – 1 after a vertical stretch by a factor of 2. The vertex of g is at (0, –2).
f(x) g(x)
Holt Algebra 2
2-9 Absolute–Value Functions
Substitute 2 for b.
Stretch the graph. f(x) = |4x| – 3 horizontally by a factor of 2.
g(x) = |2x| – 3 Simplify.
g(x) = f( x)
g(x) = | (4x)| – 3
The graph of g(x) = |2x| – 3 the graph of f(x) = |4x| – 3 after a horizontal stretch by a factor of 2. The vertex of g is at (0, –3).
g f
Holt Algebra 2
2-9 Absolute–Value Functions
Lesson Quiz: Part I
1. Translate f(x) = |x| 3 units right.Perform each transformation. Then graph.
g(x)=|x – 3|g
f
Holt Algebra 2
2-9 Absolute–Value Functions
Lesson Quiz: Part II
Perform each transformation. Then graph.
g(x)=|x – 2| – 1
2. Translate f(x) = |x| so the vertex is at (2, –1). Then graph.
f
g
Holt Algebra 2
2-9 Absolute–Value Functions
Lesson Quiz: Part III
g(x)= –3|2x| + 3
3. Stretch the graph of f(x) = |2x| – 1 vertically by a factor of 3 and reflect it across the x-axis.
Perform each transformation. Then graph.
