Functions and Graphs
Functions and Graphs tests the link between algebraic rules, tables of values and the Cartesian plane. You need to plot points accurately, interpret gradients and intercepts, recognise linear and quadratic shapes, and use a graph to solve equations or read values in context.
Core rules and concepts
- Coordinates are written (x, y): x is horizontal, y is vertical. Scale axes consistently and label them.
- A linear function has the form y = mx + c, where m is the gradient and c is the y-intercept.
- Gradient = (change in y) ÷ (change in x) = (y₂ − y₁)/(x₂ − x₁). A positive gradient rises left to right; a negative gradient falls.
- To find a line from two points, calculate the gradient, then substitute one point into y = mx + c to find c.
- A quadratic graph y = ax² + bx + c is a parabola. If a > 0 it opens upwards; if a < 0 it opens downwards. The y-intercept is c.
- Solutions of simultaneous equations are the coordinates where their graphs intersect.
- Function notation f(x) means the output when the input is x.
Worked example
Find where y = 2x + 1 and y = −x + 7 intersect. At the intersection, 2x + 1 = −x + 7. Add x: 3x + 1 = 7. Subtract 1: 3x = 6. So x = 2. Substitute: y = 2(2) + 1 = 5. The intersection is (2, 5). Check with the second rule: −2 + 7 = 5.
Traps that cost marks
Reversing the gradient formula; reading the y-intercept as the x-intercept; plotting points with the axes swapped; joining a quadratic with straight line segments; and giving only the x-value at an intersection when a coordinate pair is required.