Roots (x-intercepts)
- 3 distinct roots: the curve crosses the x-axis three times
- 2 roots: the curve crosses at one point and TOUCHES (bounces off) at another — the touching point is a repeated root
- 1 root: the curve crosses once (a triple root or just one real root)
- A cubic ALWAYS has at least one real root
Turning Points
- A cubic can have 0 or 2 turning points — never exactly 1
- Local maximum: a peak where gradient goes from positive to negative
- Local minimum: a trough where gradient goes from negative to positive
- y = x³ has a point of inflection at the origin — momentarily flat but not a true turning point
This deep dive covers Key Features of Cubic Graphs 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 4 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.
Quick recall flashcards
What does the graph of y = x³ look like?
A smooth S-shaped curve.
Key features:
- Passes through the origin (0, 0)
- Rises steeply for large positive x
- Falls steeply for large negative x
- Has a point of inflection at the origin (where it flattens then curves again)
How many roots can a cubic graph have?
A cubic graph can have 1, 2 or 3 roots (x-intercepts).
- 3 distinct roots: crosses x-axis three times
- 2 roots: touches at one point and crosses at another
- 1 root: only crosses once (with a repeated root)
Cubics ALWAYS have at least one real root.