Quadratic graphs

A quadratic graph has the equation y = ax² + bx + c and produces a parabola (U-shape if a > 0, ∩-shape if a < 0). You need to draw them by completing a table of values and identify key features.

🔑

Key facts to remember

1Quadratic graphs are parabolas — symmetrical U or ∩ shapes.
2The graph crosses the y-axis where x = 0 (substitute x = 0 to find the y-intercept).
3It crosses the x-axis where y = 0 (the roots of the quadratic).
4The turning point (vertex) lies on the line of symmetry x = −b/(2a).
5If a > 0, the parabola opens upward (minimum point); if a < 0, it opens downward (maximum point).

📐

Formulas

Line of symmetry

x = −b / (2a)

For y = ax² + bx + c

✍️

Worked examples

Example 1

Complete a table of values for y = x² − 3x + 2 for x = 0 to 4, and state the roots.

Working

x = 0: y = 0 − 0 + 2 = 2
x = 1: y = 1 − 3 + 2 = 0
x = 2: y = 4 − 6 + 2 = 0
x = 3: y = 9 − 9 + 2 = 2
x = 4: y = 16 − 12 + 2 = 6
Roots (where y = 0): x = 1 and x = 2

AnswerRoots at x = 1 and x = 2

Example 2

Find the coordinates of the turning point of y = x² − 4x + 1.

Working

Line of symmetry: x = −(−4) / (2 × 1) = 4/2 = 2
y at x = 2: y = 4 − 8 + 1 = −3
Turning point is (2, −3)

Answer(2, −3)

⚠️

Common mistakes

✗Arithmetic errors when substituting negative x values (remember (−2)² = 4, not −4).

✗Drawing a V-shape instead of a smooth curve — quadratics are smooth parabolas.

✗Not plotting enough points to identify the turning point accurately.

🎯

Exam tips

✓Always substitute carefully, especially with negative values — show each substitution step.

✓Plot at least 5–7 points and join with a smooth curve, not straight lines.

Ready to test yourself on Quadratic graphs?

Get AI-marked practice questions on exactly this subtopic.

Practice this topic →

← All topics Dashboard

▶️ Watch on YouTube

Free video lessons

Click a topic to search

▶quadratic graphs GCSE maths ▶drawing parabolas GCSE ▶quadratic graph table of values GCSE ▶y = x squared GCSE graphs

Opens YouTube — pick any free GCSE video.