Absolute value graphs are linear representations of absolute value functions. These equations are always expressed within absolute value bars. Here is an example:

Taking the absolute value of a number or equation cancels out its negative signs. Even though there is a -1 in the equation above, the x-intercept would be (1,0) since anything within absolute value bars becomes positive.

When graphing absolute value equations, the shape will either be an upright or upside-down V. The point where the "V" meets represents the vertex of the graph. Let's create a graph of the absolute value function above:

From this graph, we can determine that the vertex of this equation is (1,0), the y-intercept is (0,2), and the graph opens upward.

## Breaking Down Absolute Value Functions

Let's explain the rules of absolute value functions. The parent function, or the most basic form, of an absolute value, is:

The general form of an absolute value function is as follows:

Here’s what we can learn from this form:

- The vertex of this equation is at points (h, k).
- The horizontal axis of symmetry is marked where x = h.
- The variable k determines the vertical distance from 0.
- Whether a is positive or negative determines if the graph opens up or down.

## Analyzing an Absolute Value Graph

Let's analyze the graph of an absolute value function *f* to determine its vertex, x-intercepts, and whether it opens up or down.

From this equation, we can determine that the vertex is (-4, 3). Because *a* = -2, the graph opens down. The x-intercepts are -5.5 and -3.5, and the y-intercept is -5.

## Understanding Absolute Value Graphs

The absolute value of a number represents its distance from 0. Graphing absolute value equations allows us to visually understand this concept in terms of x and y-intercepts. It also teaches us how to determine the vertex of an absolute value function and whether it will be upward or downward-facing.