Graphs of Modulus Functions

This subtopic is the drawing arm of modulus equations: the fold-up rule, applied carefully, with the marks attached to labelled features rather than artistic quality.

The fold-up rule

$y = \lvert f(x) \rvert$: draw $y = f(x)$ lightly, then reflect every below-axis portion upward. Nothing above the axis moves.

Linear inside: $y = \lvert 3x - 6 \rvert$ is a V with vertex $(2, 0)$ and $y$-intercept $(0, 6)$, note the intercept folds from $-6$ to $+6$. Quadratic inside: $y = \lvert x^2 - 2x - 3 \rvert$ folds the dip between the roots $-1$ and $3$ into a hump; the old minimum $(1, -4)$ becomes a local maximum at $(1, 4)$; the roots become touch-points.

Where the marks are

A sketch command awards B marks for: correct shape, labelled axis crossings, labelled vertex/turning points, and (for folded quadratics) the folded peak’s coordinates. Examiners can’t award what isn’t labelled, say the coordinates on the page, not in your head.

Graphs as solving machines

The exam’s favourite use of these sketches is “solve/count using your graph”:

• Solve $\lvert x^2 - 2x - 3 \rvert = 5$ graphically: draw $y = 5$ across the sketch; the intersection $x$-values are the answers.
• Count solutions of $\lvert f(x) \rvert = k$ as $k$ varies: slide the horizontal line; the count changes at the folded peak’s height, for the example above: 4 solutions for $0 < k < 4$, 3 at $k = 4$, 2 for $k > 4$. This question is designed for the graph; algebra is the slow, error-prone route.
• “Hence” solves: if part (i) was the sketch, part (ii)‘s equation almost certainly wants intersections read from it, the hence rule.

Drawing discipline that prevents errors

1. Always draw the original function lightly first, folding from memory creates phantom shapes.
2. Compute the fold images: a $y$-intercept of $-6$ becomes $+6$; a minimum of $-4$ becomes a maximum of $+4$.
3. Touch vs cross: folded roots become touch-points (the graph meets the axis without crossing); show that in the drawing.

Common mistakes

• Above-axis sections “reflected” too (only below-axis parts fold)
• Unlabelled features, shape-only sketches score a fraction
• The folded peak’s $y$-coordinate left as the original negative value
• $y = \lvert f(x) \rvert$ confused with $y = f(\lvert x \rvert)$
• Graphical questions answered algebraically against a “hence use your graph” instruction

Full topic context: Equations, Inequalities & Graphs notes · foundations: modulus functions & graphs.