Modulus graphs
The graph of y = |x| is a V-shape with its vertex at the origin, symmetric about the y-axis. The graph of y = |ax + b| is a V whose vertex is where ax + b = 0, that is x = −b/a, and the minimum value is 0.
Key idea
For y = |x − h| + k the vertex is at (h, k) and the range is y ≥ k. Any modulus graph never goes below its vertex, so y = |ax + b| = (a negative number) has no solution.
Polynomial graphs
A polynomial in factorised form crosses the x-axis at each root. A cubic y = (x − a)(x − b)(x − c) with three distinct roots crosses the x-axis three times. A repeated factor such as (x − a)² touches the axis (a turning point) rather than crossing it.
Worked example
Find the y-intercept of y = (x − 1)(x + 2)(x − 3). Put x = 0: y = (−1)(2)(−3) = 6. So the curve cuts the y-axis at (0, 6).
Counting solutions
To count solutions of |ax + b| = k, draw the V and the horizontal line y = k. If k > 0 there are two intersections, if k = 0 one, if k < 0 none.
Remember
- The y-intercept is found by setting x = 0.
- A repeated root gives a touch, not a crossing.
- y = |f(x)| reflects any part of y = f(x) below the x-axis to above it.