Lesson 107: Graphing Absolute-Value Functions

New Concept

A function whose rule has one or more absolute-value expressions is called an absolute-value function. The absolute-value parent function is $f(x) = |x|$.

What’s next

Next, you’ll see how changing simple values in the function's equation predictably moves the graph on the coordinate plane.

Property

The absolute-value parent function is $f(x) = |x|$. The graph forms a 'V' shape with a vertex at $(0, 0)$ and an axis of symmetry at $x = 0$.

Explanation

Think of absolute value as a 'distance machine.' It tells you how far a number is from zero, which is always positive. That's why the graph bounces back up from the x-axis, creating its signature 'V' shape!

Examples

For example, both $x = -3$ and $x = 3$ result in $y = 3$, since the distance from zero is the same. The vertex, or 'corner', is at $(0,0).$

Property

The graph of $f(x) = |x - h| + k$ translates the parent function. The vertex moves to $(h, k)$.

Explanation

Imagine picking up the 'V' graph and moving it around! The 'k' value lifts it up or down, while the 'h' value slides it left or right. Just look for h and k to find the new vertex location.

Examples

In $f(x) = |x - 2| + 3$, the graph shifts right 2 and up 3. The vertex is $(2, 3)$.
In $f(x) = |x + 5| - 1$, the graph shifts left 5 and down 1. The vertex is $(-5, -1)$.

Property

For $f(x) = a|x|$, the value of 'a' reflects, stretches, or compresses the graph. If $a < 0$, the graph reflects across the x-axis.

Explanation

The 'a' value is the graph's attitude adjuster! A negative 'a' flips the V-shape upside down. A big $|a|$ makes it skinnier (stretched), and a small $|a|$ between 0 and 1 makes it wider (compressed).

Examples

For $f(x) = -2|x|$, the graph is reflected across the x-axis and stretched vertically because $a=-2$.
For $f(x) = 0.5|x|$, the graph is compressed vertically, making it wider, because $|a| < 1$.

Now let's see what happens when we put a number in front. This relates to the key idea of reflections, stretches, and compressions.

Example Problem

Describe the graph of the function $f(x) = -2|x|$.