# Absolute Value Functions

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from 0 on the number line.

The absolute value parent function, written as f(x) = |x|, is defined as

To graph an absolute value function, choose several values of x and find some ordered pairs.

Plot the points on a coordinate plane and connect them.

Observe that the graph is V-shaped.

(1) The vertex of the graph is (0, 0).

(2) The axis of symmetry ( x = 0 or y -axis) is the line that divides the graph into two congruent halves.

(3) The domain is the set of all real numbers.

(4) The range is the set of al real numbers greater than or equal to 0. .

(5) The x-intercept and the y-intercept are both 0.

### Vertical Shift

To translate the absolute value function f(x) = |x| vertically, you can use the function

g(x) = f(x) + k .

When k > 0, the graph of g(x) translated k units up.

When k < 0, the graph of g(x) translated k units down.

### Horizontal Shift

To translate the absolute value function f(x) = |x| horizontally, you can use the function

g(x) = f(x + k).

When k > 0, the graph of g(x) translated k units left.

When k < 0, the graph of g(x) translated k units right.