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.

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).

Reflection across x-axis: g(x) = –f(x)

Reflection across y-axis: g(x) = f(–x)

Remember!

Vertical stretch and compression : g(x) = af(x)

Horizontal stretch and compression: g(x) = f

Remember!

The general forms for translations are Vertical:g(x) = f(x) + kHorizontal: g(x) = f(x – h)

Remember!

Ex 1A: 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

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)

g(x)

Ex 1B: Perform the transformation on f(x) = |x|. Then graph the transformed function g(x).

1 unit leftf(x) = |x|

g(x) = f(x – h )

g(x) = |x – (–1)| = |x + 1|

f(x)

g(x)

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).

Because the entire graph moves when shifted, the shift from f(x) = |x| determines the vertex of an absolute-value graph.

The graph confirms that the vertex is (–1, –3).

f(x)

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.

g(x)

Ex 2: 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

g f

The vertex of the graph g(x) = |–x – 2| + 3 is (–2, 3).

Ex 3A: 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.

g(x) = af(x)g(x) = 2(|x| – 1) Multiply the entire function by 2.

Ex 3B: 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)

Ex 3C: Compress the graph of f(x) = |x + 2| – 1 horizontally by a factor of .

g(x) = |2x + 2| – 1

Simplify.

Substitute for b.

f

The graph of g(x) = |2x + 2|– 1 is the graph of f(x) = |x + 2| – 1 after a horizontal compression by a factor of . The vertex of g is at (–1, –1).

g