Graphs of $y = |f(x)|$ and $y = f(|x|)$
Big Idea
The absolute value function can be applied to a function in two fundamentally different ways — either to the output or to the input. Each produces a dramatically different transformation of the original graph. Mastering these two transformations gives you powerful tools for sketching complex graphs quickly.
📌 Quick Review: Absolute Value
Recall that the absolute value of a number $a$ is defined as:
$$ |a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases} $$In simple terms: absolute value makes negative values positive and leaves positive values unchanged.