From a Factorised Form
If given y = (x − a)(x − b)(x − c):
- Find roots: set each bracket to zero → x = a, x = b, x = c
- Find y-intercept: set x = 0, multiply all brackets
- Identify the shape: positive leading coefficient → positive cubic shape
- Plot the roots and y-intercept, sketch a smooth S-shaped curve through them
Example: Sketch y = (x + 1)(x − 2)(x − 3)
Roots: x = −1, x = 2, x = 3 (set each bracket to zero)
y-intercept: (0 + 1)(0 − 2)(0 − 3) = 1 × (−2) × (−3) = 6 → point (0, 6)
Leading term is x³ (positive), so positive cubic shape.
Sketch: rises from bottom-left, crosses at x = −1, dips below x-axis, crosses at x = 2, rises above, crosses at x = 3, continues upward.
From a Table of Values
- Choose a range of x-values (e.g. −3 to 3)
- Substitute each value into the equation to calculate y
- Plot all points carefully
- Join with a smooth S-shaped curve — no sharp corners or straight segments
This deep dive covers Drawing a Cubic Graph within Cubic Graphs for GCSE Mathematics. Revise Cubic Graphs in Graphs for GCSE Mathematics with 11 exam-style questions and 10 flashcards. This topic appears less often, but it can still be a useful differentiator on mixed-topic papers. It is section 5 of 10 in this topic. Use this deep dive to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Cubic Graphs
Which of the following best describes the general shape of the graph y = x³?
Explain how you can tell from the equation of a cubic whether its graph rises or falls as x approaches positive infinity.