Functions and Graphs
What this domain tests
Reading and drawing proportional (y = ax), linear (y = ax + b), inverse-proportion (y = a/x) and quadratic (y = ax²) graphs, and using a graph to solve an equation.
Core rules
- In y = ax + b, a is the slope and b the y-intercept. Slope = rise/run = (y₂−y₁)/(x₂−x₁).
- y = a/x is a hyperbola with the axes as asymptotes and x ≠ 0. When a > 0 the branches lie in quadrants I and III; when a < 0 they lie in II and IV.
- y = ax² has its vertex at the origin and axis x = 0. Positive a opens upward; a larger |a| makes the curve narrower.
- Intersections are found by setting the two expressions equal; the x-values solve the resulting equation, and the y-values come from substituting back.
- The rate of change of y = ax² varies along the curve, which is exactly what separates it from a straight line.
Worked example
Find where y = x² meets y = 2x + 1. Set x² = 2x + 1, so x² − 2x − 1 = 0, giving x = 1 ± √2. Then y = 2(1 ± √2) + 1 = 3 ± 2√2. There are two intersections, one of them at a negative x-value.
Traps that cost marks
Reading an intercept as the slope; expecting a line and a parabola to meet exactly once (there may be two points or none, depending on the discriminant); forgetting x ≠ 0 in inverse proportion; plotting too few points to reveal the shape.