Sheet № 27 · Higher only · AQA · Edexcel · OCR
Cubic, Reciprocal & Exponential Graphs –
At Higher tier GCSE Maths, you are expected to recognise and sketch several types of graph beyond straight lines and quadratics. Cubic, reciprocal and exponential graphs each have distinctive shapes and key features that examiners love to test. This guide covers what each graph looks like, how to plot them from a table of values, and how
§Key definitions
Question:
Sketch the graph of y = x³ − 3x for −3 ≤ x ≤ 3.
Question (a):
Sketch y = 3/x and label the asymptotes.
Question (b):
Sketch y = 2ˣ for −3 ≤ x ≤ 4.
Question 1:
Match each equation to its graph type: (a) y = 5/x, (b) y = x³ + 2, (c) y = 4ˣ.
Question 2:
Complete a table of values for y = 2/x for x = −4, −2, −1, 1, 2, 4 and describe the shape of the graph.
§Formulas to memorise
y = ax³ + bx² + cx + d
y = a/x (equivalently y = ax⁻¹)
y = aˣ (where a > 0)
Two asymptotes — the x-axis (y = 0) and the y-axis (x = 0).
It may have two turning points (a local maximum and a local minimum), or none for y = x³ itself.
The curve passes through (0, 1) because a⁰ = 1 for any a.
The x-axis (y = 0) is a horizontal asymptote — the curve approaches it but never touches it.
Look at the highest power of x — or where x appears.
If the highest power is x³ → cubic.
If x is in the denominator (1/x, a/x) → reciprocal.
If x is in the exponent (2ˣ, 3ˣ) → exponential.
Worked example
Sketch the graph of y = x³ − 3x for −3 ≤ x ≤ 3.
Table of values:
⚠ Common mistakes
- ✗Drawing straight lines between points on a cubic. All these curves must be smooth. Use a flowing motion with your pencil.
- ✗Forgetting asymptotes on reciprocal graphs. Always draw them as dashed lines and label them. This is worth marks on its own.
- ✗Thinking y = 2ˣ passes through the origin. It does not — it passes through (0, 1), because 2⁰ = 1.
- ✗Confusing x² and x³ shapes. A quadratic is U-shaped and symmetric; a cubic has an S-shape and is not symmetric.
- ✗Plotting y = 1/x at x = 0. The function is undefined at x = 0. Leave a gap and do not let the curve cross the y-axis.
✦ Exam tips
- →Learn the shapes. In matching questions, you need to pair equations to graphs instantly. Practise until you can recognise cubic, reciprocal and exponential shapes at a glance.
- →Mark key points. For exponential graphs, always plot (0, 1). For reciprocal graphs, plot (1, a) and (−1, −a). These quick reference points help you draw accurately.
- →Use a ruler for asymptotes but a freehand smooth curve for the graph itself.
- →On Edexcel papers, you may be given a graph and asked which equation it matches. Check the shape first, then verify with one or two coordinates.