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Quadratic Graphs: Roots, Turning Points and Symmetry

Sketch parabolas and find the roots, y-intercept, line of symmetry and turning point (vertex) of a quadratic, with step-by-step worked examples.

  • ⏱ 3 min read
  • ✏️ 3 worked examples
  • 🎯 5 practice questions
  • 📶 Intermediate
GCSEA-LevelHigh School
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Curriculum fit: GCSE Maths Algebra (Foundation and Higher). US: Algebra I and II quadratic functions (Interpreting Functions).

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What Is It?

The graph of a quadratic function y = ax² + bx + c is a smooth, symmetrical curve called a parabola. If a is positive the curve is U-shaped and has a minimum point. If a is negative it is ∩-shaped and has a maximum point.

Key features to find are the roots (where the curve crosses the x-axis), the y-intercept, the line of symmetry and the turning point (the vertex).

y = x² − 2x − 3roots: (−1, 0) and (3, 0)y-intercept: (0, −3)turning point: (1, −4)line of symmetry: x = 1
A quadratic graph is a parabola. It is symmetrical about a vertical line through the turning point, and crosses the x-axis at its roots.

Key Formulas and Ideas

  • Roots: solve ax² + bx + c = 0.
  • y-intercept: the value of c (put x = 0).
  • Axis of symmetry: x = −b ÷ (2a).
  • Turning point: substitute the axis of symmetry value into the equation to find y.

Worked Examples

Example 1

Roots and y-Intercept

Question

For y = x² − 4x + 3, find the roots and y-intercept.

  1. Set y = 0: x² − 4x + 3 = 0.
  2. Factorise: (x − 1)(x − 3) = 0, so x = 1 or x = 3.
  3. y-intercept: x = 0 gives y = 3.

✓ Answer Roots 1 and 3; y-intercept 3

Example 2

Turning Point

Question

Find the turning point of y = x² − 4x + 3.

  1. Axis of symmetry: x = −(−4) ÷ (2 × 1) = 2.
  2. Substitute x = 2: y = 4 − 8 + 3 = −1.

✓ Answer (2, −1), a minimum

Example 3

Shape

Question

Describe the shape of y = −x² + 6x.

  1. a = −1 is negative, so the curve is ∩-shaped.
  2. Roots: −x(x − 6) = 0, so x = 0 or x = 6.
  3. Axis x = 3; y = −9 + 18 = 9.

✓ Answer ∩-shaped with maximum point (3, 9)

Common Mistakes to Avoid

  • Joining the plotted points with straight lines instead of a smooth curve.
  • Confusing the roots with the y-intercept.
  • Forgetting the minus sign in −b ÷ 2a.
  • Giving only the x-value of the turning point rather than both coordinates.

Practice Questions

0 of 5 answered
Q1A quadratic has a positive x² coefficient. Its graph is…
Q2What is the y-intercept of y = x² + 5x − 7?
Q3What are the roots of y = x² − 9?
Q4Find the axis of symmetry of y = x² + 6x + 1.
Q5A parabola has roots 2 and 6. Where is its axis of symmetry?

Frequently Asked Questions

Can a quadratic have no roots?

Yes. If the curve never reaches the x-axis it has no real roots.

How are the roots related to the line of symmetry?

The axis of symmetry is exactly halfway between the two roots.

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Sources and Review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.