Sketch graphs involving absolute value and solve absolute value equations and inequalities.

What this dot point is asking

This dot point develops fluency with the absolute value (modulus) function, both as a graph to sketch and as something to solve. The single defining idea is that $|x|$ measures size without regard to sign.

Sketching the basic graph

The graph of $y = |x|$ is a V shape with its vertex at the origin, made of the line $y = x$ for $x \ge 0$ and the line $y = -x$ for $x < 0$. Transformations work just like any function: $y = |x - 2| + 1$ shifts the V two units right and one unit up, so its vertex sits at $(2, 1)$.

Sketching y equals the modulus of f of x

This is different from $y = f(|x|)$, which instead takes the right-hand part of $f$ and reflects it across the $y$ axis to make an even function. Read carefully which version a question wants.

Solving absolute value equations

To solve $|f(x)| = a$ (with $a \ge 0$), the inside can be either $+a$ or $-a$:

Always check solutions in the original equation, and reject any if $a < 0$ since an absolute value can never be negative.

Solving absolute value inequalities

Two standard patterns cover most cases:

The first traps the inside between $-a$ and $a$; the second lets the inside escape on either side.

Why this matters

Absolute value graphs appear in piecewise modelling and in defining regions, and the case-split logic here is the same reasoning used for reciprocal and rational graphs. Mastering the reflection rule and the two inequality patterns makes those later graphs routine.