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Key Features of Cubic Graphs

Part of Cubic Graphs — GCSE Mathematics

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.

Topic position

Section 4 of 10

Practice

11 questions

Recall

10 flashcards

Key Features of Cubic Graphs

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

Keep building this topic

Read this section alongside the surrounding pages in Cubic Graphs. That gives you the full topic sequence instead of a single isolated revision point.

Positive vs Negative Cubic ShapesDrawing a Cubic GraphThe Roller Coaster CurveCubic Graph Essentials

Practice Questions for Cubic Graphs

Which of the following best describes the general shape of the graph y = x³?

  • A. U-shape (parabola) opening upward
  • B. S-shaped curve rising from bottom-left to top-right
  • C. Horizontal straight line
  • D. S-shaped curve falling from top-left to bottom-right
1 markfoundation

Explain how you can tell from the equation of a cubic whether its graph rises or falls as x approaches positive infinity.

2 markshigher

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.

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Positive vs Negative Cubic Shapes

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Drawing a Cubic Graph