What is important to know about Quadratic & cubic graphs?

Quadratic (y = ax² + bx + c): parabola; 0, 1 or 2 roots; one turning point.

What is important to know about Quadratic & cubic graphs?

Cubic (y = ax³ + bx² + cx + d): S-shaped curve; up to 3 roots; up to 2 turning points.

What is important to know about Quadratic & cubic graphs?

Positive cubic (a > 0): goes from bottom-left to top-right.

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Algebra · Higher

Quadratic & cubic graphs

Q: What are common mistakes with Quadratic & cubic graphs?

Drawing a cubic as a simple curve without the S-shape characteristic.

Q: What are common mistakes with Quadratic & cubic graphs?

Missing the y-intercept when sketching.

Q: What exam tips are there for Quadratic & cubic graphs?

For sketches: label all intercepts with coordinates, mark the turning point, and show the correct general shape.

Q: What exam tips are there for Quadratic & cubic graphs?

Use factorisation to find roots rather than plotting individual points for sketch questions.

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

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 2

For 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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