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Algebra · Higher

Quadratic & cubic graphs (extended)

Quadratic graphs (y = ax² + bx + c) are U-shaped (or ∩-shaped) parabolas. Cubic graphs (y = ax³ + bx² + cx + d) have a characteristic S-shape. You need to sketch, interpret and read values from these graphs.

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Revising quadratic & cubic graphs (extended) 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 (extended) GCSE” on the site for the complete guide.

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Key facts to remember

  • 1Quadratic y = ax² + bx + c: positive a → U-shape, negative a → ∩-shape.
  • 2The roots (x-intercepts) are found by solving the equation = 0.
  • 3The turning point (vertex) lies on the line of symmetry x = −b/(2a).
  • 4The y-intercept of a quadratic is always c (set x = 0).
  • 5Cubic y = ax³: positive a → bottom-left to top-right, negative a → top-left to bottom-right.
  • 6Cubic graphs can cross the x-axis up to 3 times.
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Formulas

Line of symmetry (quadratic)
x = −b / (2a)
y-intercept
Set x = 0: y = c
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Worked examples

Example 1

Sketch the graph of y = x² − 4x + 3, marking the roots and turning point.

Working

  1. Find roots: x² − 4x + 3 = 0 → (x − 1)(x − 3) = 0 → x = 1 or x = 3.
  2. Find line of symmetry: x = −(−4)/(2×1) = 2.
  3. Find turning point: y = (2)² − 4(2) + 3 = 4 − 8 + 3 = −1. Vertex: (2, −1).
  4. y-intercept: x = 0 → y = 3. Point (0, 3).
  5. Sketch U-shape crossing x-axis at x = 1 and x = 3, vertex at (2, −1).
AnswerU-shaped parabola with roots at (1, 0) and (3, 0), vertex at (2, −1), y-intercept at (0, 3).
Example 2

Sketch y = x³ − 3x, stating the coordinates where it crosses the axes.

Working

  1. Find x-intercepts: x³ − 3x = 0 → x(x² − 3) = 0 → x = 0, x = √3, x = −√3.
  2. y-intercept: x = 0 → y = 0. Passes through origin.
  3. Positive cubic: bottom-left to top-right with S-shape.
  4. Mark (0,0), (√3, 0) ≈ (1.73, 0), (−√3, 0) ≈ (−1.73, 0).
AnswerS-shaped cubic crossing at x = −√3, x = 0, x = √3.
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Common mistakes

✗Drawing a quadratic as a V-shape instead of a smooth curve.
✗Forgetting that a negative leading coefficient flips the shape.
✗Confusing the line of symmetry formula — it is −b/2a not b/2a.
✗Drawing a cubic that doesn't extend in both directions indefinitely.
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Exam tips

✓Always find and label the roots, y-intercept and turning point when sketching.
✓Check whether a is positive or negative to determine the shape.
✓Use a table of values if you're unsure of the shape.
✓For cubics, plug in a few x-values to check your sketch is correct.

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