Modulus Functions & Their Graphs

The modulus $\lvert x \rvert$ strips the sign: $\lvert 5 \rvert = 5$ and $\lvert -5 \rvert = 5$. Graphically, $y = \lvert f(x) \rvert$ is $y = f(x)$ with every below-axis section reflected upward, the “fold-up” rule. In 0606 this subtopic is mostly examined through sketches, and sketch marks live in the labels.

Sketching $y = \lvert ax + b \rvert$

Sketch $y = \lvert 2x - 5 \rvert$.

1. Sketch $y = 2x - 5$ lightly: crosses the $x$-axis at $x = \frac{5}{2}$, the $y$-axis at $-5$.
2. Fold the below-axis portion up.
3. Label: vertex $(\frac{5}{2}, 0)$ and $y$-intercept $(0, 5)$, the folded intercept becomes positive.

The V-shape with both features labelled is typically 2 B marks. Unlabelled, the same drawing can score zero, sketch commands are feature commands.

Sketching $y = \lvert \text{quadratic} \rvert$ and $y = \lvert f(x) \rvert$ generally

Same fold-up rule: draw the original, reflect the negative arcs. For $y = \lvert x^2 - 4 \rvert$: the parabola’s dip between $x = -2$ and $2$ folds up into a hump touching $(\pm 2, 0)$, with a local maximum at $(0, 4)$. Mark the axis crossings (now touch-points) and the folded peak.

Using the graph to solve and count

Modulus graphs turn equations into intersections:

• Solve $\lvert 2x - 5 \rvert = 3$: draw $y = \lvert 2x - 5 \rvert$ and $y = 3$; two intersections $\to$ two solutions ($x = 1, x = 4$). Algebraic route in modulus equations.
• “State the number of solutions of $\lvert x^2 - 4 \rvert = k$”: read it straight off the sketch as the horizontal line $y = k$ slides, e.g. 4 solutions for $0 < k < 4$, 3 at $k = 4$, 2 for $k > 4$. This count-by-graph question is a 0606 favourite and is designed to be answered from the picture, not algebra.

Common mistakes

• Folding the wrong part (reflect only what’s below the axis, the above-axis part doesn’t move)
• Vertex/intercept coordinates missing or unconverted (the $y$-intercept of $y = \lvert 2x - 5 \rvert$ is $+5$)
• $y = \lvert f(x) \rvert$ confused with $y = f(\lvert x \rvert)$, the latter mirrors the right half leftward instead; 0606 focuses on $\lvert f(x) \rvert$
• Solution counts attempted algebraically when the sketch answers in seconds

Full topic context: Functions notes · the equation-solving side lives in Equations, Inequalities & Graphs.