2-9 Absolute–Value Functions
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 )

2-9 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 (___, ___)

2-9 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.

2-9 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

2-9 Absolute–Value Functions
Stretch the graph. f(x) = |x| – 1 vertically by a factor of 2.

2-9 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.

2-9 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.