Relations, Functions and Graphs
What you must understand
This domain connects equations to pictures. You must plot and read straight-line graphs, find gradient and intercept from y = mx + c, determine the equation of a line from two points or from a graph, interpret distance–time and velocity–time graphs, draw and read quadratic, cubic and reciprocal curves, use function notation, find inverse and composite functions, and solve equations graphically by reading intersections.
Traps that appear in questions
- Gradient sign. A line falling from left to right has a negative gradient.
- Gradient formula. m = (y₂ − y₁) ÷ (x₂ − x₁); subtract in the same order both times.
- Parallel and perpendicular. Parallel lines share m; perpendicular gradients multiply to −1.
- Graph interpretation. On a distance–time graph the gradient is speed; on a velocity–time graph the gradient is acceleration and the area is distance.
- Function notation. f⁻¹ reverses the operations in the opposite order.
- Graphical solutions. The solution of simultaneous equations is the point of intersection, read from both coordinates.
Worked example
Find the equation of the line through (1, 5) and (4, 14). Gradient m = (14 − 5) ÷ (4 − 1) = 9 ÷ 3 = 3. Using y = mx + c at (1, 5): 5 = 3(1) + c, so c = 2. The equation is y = 3x + 2. Check with (4, 14): 3(4) + 2 = 14.
Rule of thumb
For any straight line, find the gradient first, then substitute one known point to find the intercept - and always check the finished equation against the second point.