Revising quadratic & cubic graphs the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Quadratic & cubic graphs GCSE” on the site for the complete guide.
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Key facts to remember
- 1Quadratic (y = ax² + bx + c): parabola; 0, 1 or 2 roots; one turning point.
- 2Cubic (y = ax³ + bx² + cx + d): S-shaped curve; up to 3 roots; up to 2 turning points.
- 3Positive cubic (a > 0): goes from bottom-left to top-right.
- 4Negative cubic (a < 0): goes from top-left to bottom-right.
- 5The y-intercept is always found by setting x = 0.
- 6Roots (x-intercepts) are found by setting y = 0.
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Worked examples
Example 1Sketch y = x³ − 4x, labelling all intercepts.
Working
- y-intercept: x = 0 → y = 0
- x-intercepts: x³ − 4x = 0 → x(x² − 4) = 0 → x(x+2)(x−2) = 0
- Roots at x = −2, 0, 2
- Positive cubic: rises from bottom-left to top-right, crossing x-axis at −2, 0 and 2
AnswerCubic crossing x-axis at x = −2, 0 and 2, y-intercept at origin
Example 2For y = x² − 5x + 4, find the roots and turning point.
Working
- Factorise: (x − 1)(x − 4) = 0 → roots x = 1 and x = 4
- Line of symmetry: x = (1 + 4)/2 = 2.5
- y at x = 2.5: 6.25 − 12.5 + 4 = −2.25
- Turning point: (2.5, −2.25)
AnswerRoots x = 1 and x = 4; turning point (2.5, −2.25)
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Common mistakes
✗Drawing a cubic as a simple curve without the S-shape characteristic.
✗Missing the y-intercept when sketching.
✗Confusing negative quadratic (∩ shape) with positive quadratic (U shape).
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Exam tips
✓For sketches: label all intercepts with coordinates, mark the turning point, and show the correct general shape.
✓Use factorisation to find roots rather than plotting individual points for sketch questions.
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