Functions and Graphs
What the domain tests
This domain asks you to read information off a graph, to draw a graph from a formula or a table, and to interpret a graph that models a real situation. The functions in the basic-school programme are the linear function y = kx + b, direct proportionality y = kx, and the quadratic function y = ax² + bx + c.
Core rules
- Linear function: the graph is a straight line. The gradient k = Δy/Δx = (y₂ − y₁)/(x₂ − x₁) measures change in y per unit change in x; b is the y-intercept, so the line crosses the y-axis at (0, b). If k > 0 the function increases, if k < 0 it decreases, if k = 0 it is constant. Parallel lines share the same k. The zero is found from kx + b = 0, giving x = −b/k.
- Proportionality: a special case with b = 0, so the line passes through the origin and k = y/x. Doubling x doubles y.
- Quadratic function: the graph is a parabola. If a > 0 it opens upwards and the vertex is a minimum; if a < 0 it opens downwards and the vertex is a maximum. The axis of symmetry is x = −b/(2a), the y-intercept is c, and the zeros are where the graph meets the x-axis.
- Reading graphs: give the domain and range, the intervals where the function increases or decreases, the maximum or minimum, and the x-values for which f(x) > 0.
- Real contexts: the gradient is a rate (for example km/h or euros per item), and both axes carry units.
Worked example
A straight line passes through (2, 5) and (6, 13). Find its equation and its zero. k = (13 − 5)/(6 − 2) = 8/4 = 2. Use one point in y = kx + b: 5 = 2·2 + b, so b = 1. The equation is y = 2x + 1. Its zero satisfies 2x + 1 = 0, so x = −0,5, and the line crosses the y-axis at (0, 1).
Traps that cost marks
- Swapping the gradient and the intercept, or writing the intercept as (b, 0) instead of (0, b).
- Reversing Δy and Δx when calculating k.
- Ignoring the scale on the axes - one square may be 2 or 5 units, not 1.
- Sign slips in x = −b/(2a) for the axis of symmetry of a parabola.
- Assuming a relationship is linear when the graph is actually curved, or quoting an answer outside the realistic domain (for example a negative time).