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Algebra · Higher

Quadratic & cubic graphs

At Higher tier you need to sketch and interpret quadratic, cubic and other polynomial graphs, identifying key features such as roots, turning points and intercepts.

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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 1

Sketch y = x³ − 4x, labelling all intercepts.

Working

  1. y-intercept: x = 0 → y = 0
  2. x-intercepts: x³ − 4x = 0 → x(x² − 4) = 0 → x(x+2)(x−2) = 0
  3. Roots at x = −2, 0, 2
  4. 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 2

For y = x² − 5x + 4, find the roots and turning point.

Working

  1. Factorise: (x − 1)(x − 4) = 0 → roots x = 1 and x = 4
  2. Line of symmetry: x = (1 + 4)/2 = 2.5
  3. y at x = 2.5: 6.25 − 12.5 + 4 = −2.25
  4. 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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