Key Idea: This topic draws three new graphs from a given graph of f, and solves modulus equations and inequalities. It is on Paper 1 and Paper 2, often in the long Paper 2 function question.
| Graph | What changes |
|---|---|
| |f(x)| | Parts below the x-axis flip up |
| f(|x|) | Keep x ≥ 0, mirror it to x < 0 |
| 1/f(x) | Zeros become vertical asymptotes; max ↔ min |
Modulus equations & inequalities
- |inside| = c: inside = +c or −c (need c ≥ 0).
- |inside| < a: a band, −a < inside < a.
- |inside| > a: outside, inside < −a or inside > a.
- |f| = |g|: f = g or f = −g.
Mark intercepts, turning points and asymptotes. They move in a predictable way.
y = |x² − 4| between −2 and 2? The dip flips up into a hump.
Solve |x − 4| = |x + 2|. x − 4 = −(x + 2), so x = 1.
Solve |x| < 3. −3 < x < 3.
f has a zero at x = 2. What does 1/f have there? A vertical asymptote.