This topic summary covers Knowledge Organiser: 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 10 of 10 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Knowledge Organiser: Cubic Graphs
Key Facts
- General form: y = ax³ + bx² + cx + d
- a > 0: rises bottom-left to top-right
- a < 0: falls top-left to bottom-right
- Always at least 1 real root
- Has 0 or 2 turning points (never 1)
Sketching Steps (Factorised Form)
- 1. Find roots: set each bracket to zero
- 2. Find y-intercept: substitute x = 0
- 3. Check sign of a (shape direction)
- 4. Sketch smooth curve through roots and y-intercept
Key Vocabulary
- Cubic: polynomial of degree 3
- Point of inflection: momentarily flat, no turning point
- Repeated root: curve touches axis without crossing
- Turning point: local max or min of the curve
Common Errors
- Drawing straight line segments instead of a smooth curve
- Making a cubic cross at a repeated root (should touch)
- Assuming all cubics have an S-shape (many have a hill-valley shape)
- Thinking cubics can only cross x-axis once
Key Formulas
- General form: y = ax³ + bx² + cx + d
- Positive a: starts bottom-left, ends top-right
- Negative a: starts top-left, ends bottom-right
- x-intercepts: set y = 0 and solve (can have 1, 2, or 3 roots)
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.