Step 1: Identify the function family
Parabola, hyperbola, exponential or log — the family decides which features matter.
Step 2: Find key features
- x- and y-intercepts
- Turning point or asymptotes
- Axis of symmetry where relevant
Step 3: Sketch and label
Every labelled feature earns a mark.
Worked example: a parabola
y = −x² + 2x + 3. The y-intercept is 3. Factorising −(x − 3)(x + 1) = 0 gives x-intercepts at x = 3 and x = −1. The turning point is at x = 1, where y = 4, so it is (1; 4).
Worked example: a hyperbola
y = 2/(x − 1) + 3. The asymptotes are x = 1 and y = 3. The y-intercept is y = −2 + 3 = 1. For the x-intercept, 2/(x − 1) = −3, so x − 1 = −⅔ and x = ⅓.
Finding the equation from a graph
Choose the form that uses what the graph shows. If the turning point is given, use y = a(x − p)² + q and substitute another point to find a. If the x-intercepts are given, use y = a(x − x₁)(x − x₂). For a hyperbola, read p and q from the asymptotes, then substitute a point to find a. Example: a parabola has x-intercepts −1 and 3 and passes through (0; −3). Then y = a(x + 1)(x − 3), and −3 = a(1)(−3) gives a = 1, so y = x² − 2x − 3.
Exponential graphs
For y = a·bˣ + q, the asymptote is y = q and the y-intercept is a + q. Example: h(x) = 3·2ˣ − 6 has asymptote y = −6, y-intercept −3, and its x-intercept is where 2ˣ = 2, so x = 1.
Inverse functions
- Swap x and y, then make y the subject.
- The inverse of y = bˣ is y = log_b x.
- A function and its inverse are reflections of each other in the line y = x.
- For y = ax², restrict the domain to x ≥ 0 or x ≤ 0 so that the inverse is a function.
Reading answers off a graph
- f(x) > 0: where the graph is above the x-axis.
- f(x) = g(x): the x-values of the intersection points.
- f(x) ≥ g(x): where f lies on or above g.
- Average gradient between two points: the change in y divided by the change in x.
Step 4: Answer the actual question asked
Inverse? Average gradient? Range? Re-read the verb before writing. When a question stops you, snap it into Snap&Learn — you'll get a CAPS-aligned step-by-step explanation in seconds, and your first three AI solutions are free.