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The Roller Coaster Curve
The Roller Coaster Curve
A roller coaster rises, reaches a peak, descends into a valley, then climbs again before plummeting off the end. That distinctive shape — a hill followed by a valley — is exactly what many cubic graphs look like. Unlike the simple U-shape of a quadratic, a cubic curve can change direction twice, cross the x-axis up to three times, and has a characteristic S-shape that appears everywhere from physics equations to economic models.
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)
Mathematics glossary
- 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)
Earn the mark scheme marks
💡 Exam Tips for Cubic Graphs
- Smooth curve: always join points with a smooth S-shaped curve — never with straight segments or sharp corners
- Check the shape first: positive leading coefficient → bottom-left to top-right; negative → top-left to bottom-right
- Repeated roots: at a repeated root the curve TOUCHES the x-axis but does not cross it (like a "bounce")
- Point of inflection: at a point of inflection the curve flattens momentarily but does not change direction — gradient does not equal zero at a true inflection
- Always calculate the y-intercept: set x = 0 and mark this point — it anchors your sketch
Now try it yourself
Quiz · Question 1 of 11
Which of the following best describes the general shape of the graph y = x³?
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