Summary — What This Chapter Covers
- Modulus function — makes any input positive.
|7| = 7 and |-7| = 7.
- Sketching modulus graphs — draw the original graph, then reflect anything below the x-axis upward.
- Solving modulus equations — sketch both sides, find intersections, work out which "piece" (normal or reflected) applies at each one.
- Solving modulus inequalities — same sketching approach, but shade the region satisfying the inequality instead of just finding points.
- Cubic polynomials — degree-3 functions, always in factorised form for sketching, four possible shapes depending on the sign of
a and the roots.
- Modulus cubic graphs — apply the same reflection rule to cubics; the graph will only touch (never cross) the x-axis.
- Finding a cubic from its modulus graph — read off roots and the y-intercept, then solve for the leading coefficient.
- Solving cubic inequalities graphically — rearrange to zero, factorise, sketch, then read off the x-values where the curve sits above/below the axis.
1. Modulus Functions
1.1 What is a modulus function?
Think of the modulus (also called absolute value) as a machine that strips away the sign of whatever
goes in and just tells you the size — the distance from zero. It doesn't care whether the number was positive
or negative to begin with; it only cares "how far from zero is this?"
It's written with a pair of vertical bars around the input — similar to how brackets group things together:
Two forms you'll meet in this course:
Quadratic form
|ax² + bx + c|- e.g.
|x² + x − 6|
1.2 Sketching y = |f(x)|
Here's the trick that makes this whole topic click: you never need to sketch a modulus graph from scratch.
You sketch the ordinary graph first (the "before" picture), and then apply one simple rule.
The Two-Step Method
1 Pencil in the graph of y = f(x) as normal.
2 Reflect anything below the x-axis upward, in the x-axis, to get y = |f(x)|.
y y
│ / │ /
│ / y=f(x) │ / y=f(x)
│ / │ /
───┼───●─────── x ──► ───┼───●─────── x
│ / │ ╲ (reflected up)
│ / (this dips │ ╲ y=|f(x)|
│/ below axis) │ ╲
Two things to notice
1. If the original y-intercept was negative, it becomes positive on the modulus graph.
2. At every x-axis crossing, the modulus graph gets a sharp "V" shape (not a smooth curve like a turning point) — because reflecting creates a corner, not a curve.
Watch out
If the graph is y = −|f(x)| instead, the whole thing is now wholly negative — every crossing point becomes a sharp "Λ" (upside-down V) instead. Don't mix these two up!
Practice Question
Sketch y = |x − 4|. State the coordinates of the vertex and the y-intercept.
1.3 Solving modulus equations
more than one solution
y = x + 1
y = 2x
once
y = |2x|
y = |x + 1|
twice
Method: Solving |f(x)| = |g(x)| type equations
1 Sketch both graphs, including any reflected (modulus) parts.
2
3