Quadratic Graphs GCSE – Plotting & Key Features

Quadratic graphs appear on every GCSE Maths paper and are worth significant marks at both Foundation and Higher tier. You need to recognise the characteristic U-shape (or inverted U-shape), plot quadratics from a table of values, and identify key features such as roots, turning points and lines of symmetry. At Higher level, you must also sketch quadratics from their equation and use completing the square to find the turning point. This guide covers everything you need.

What Is a Quadratic Graph?

A quadratic graph is the curve produced when you plot a quadratic function — any function where the highest power of x is x². The general form is:

y = ax² + bx + c

The shape of a quadratic graph is called a parabola.

If a > 0 (positive x²), the parabola is U-shaped and has a minimum turning point.
If a < 0 (negative x²), the parabola is ∩-shaped and has a maximum turning point.

Key features

Roots (or solutions): the x-values where the curve crosses the x-axis (where y = 0). A quadratic can have 0, 1 or 2 roots.
y-intercept: the point where the curve crosses the y-axis (always the value of c).
Turning point (vertex): the minimum or maximum point of the curve.
Line of symmetry: a vertical line through the turning point. For y = ax² + bx + c:

x = −b / (2a)

Turning point form

When a quadratic is written in completed square form:

y = a(x − p)² + q

the turning point is at (p, q).

Step-by-Step Method

How to plot a quadratic graph

Draw a table of values. Choose x-values that cover the range given in the question (typically −3 to 3 or −2 to 5).
Substitute each x-value into the equation to calculate y.
Plot the points carefully on the grid.
Join the points with a smooth curve — never use straight-line segments between points.
Label the curve with its equation.

How to identify key features from a graph

Roots: read off where the curve crosses the x-axis.
Turning point: identify the lowest (or highest) point of the curve.
Line of symmetry: draw a vertical line through the turning point.

How to find the turning point algebraically (Higher)

Complete the square on y = ax² + bx + c to write it as y = a(x − p)² + q.
Read off (p, q) — that is the turning point.

Worked Example 1 — Foundation Level

Question: Complete a table of values for y = x² − 4x + 3 for x = 0 to 5, then sketch the graph and state the roots and turning point.

Table of values:

x	0	1	2	3	4	5
y	3	0	−1	0	3	8

Calculations: When x = 0, y = 0 − 0 + 3 = 3. When x = 2, y = 4 − 8 + 3 = −1.

Roots: The curve crosses the x-axis at x = 1 and x = 3 (where y = 0).

Turning point: The lowest y-value is −1, at x = 2. So the turning point is (2, −1).

Line of symmetry: x = 2.

Worked Example 2 — Higher Level

Question: By completing the square, find the turning point of y = 2x² − 12x + 22.

Step 1: Factor out the coefficient of x² from the first two terms:

y = 2(x² − 6x) + 22

Step 2: Complete the square inside the bracket. Half of −6 is −3, and (−3)² = 9:

y = 2[(x − 3)² − 9] + 22

Step 3: Expand and simplify:

y = 2(x − 3)² − 18 + 22

y = 2(x − 3)² + 4

Step 4: Read off the turning point from y = a(x − p)² + q:

The turning point is (3, 4).

Since a = 2 > 0, this is a minimum point.

Common Mistakes

Joining points with straight lines. A quadratic graph is a smooth curve, not a series of straight segments. Examiners will penalise jagged lines.
Plotting errors from the table. Double-check your substitutions, especially when squaring negative numbers: (−3)² = 9, not −9.
Confusing roots with turning points. Roots are where y = 0 (on the x-axis). The turning point is the minimum or maximum of the curve — it may not be on the x-axis at all.
Sign errors in completing the square. When you write y = a(x − p)² + q, the turning point x-coordinate is +p, not −p. For example, y = (x − 3)² + 1 has turning point (3, 1), not (−3, 1).
Forgetting to multiply back. When factoring out 'a' before completing the square, remember to multiply the constant you create back by 'a'.

Exam Tips

Use a sharp pencil for plotting and drawing curves — thick lines make it hard for the examiner to see exactly where the curve passes.
Read values from the turning point carefully. If a question asks for the minimum value of y, the answer is the y-coordinate of the turning point.
On AQA and OCR papers, you may be asked to "estimate the solutions" from a graph. Give values to 1 decimal place and look for where the curve crosses the x-axis.
If the question says "sketch", you do not need a table of values — just show the correct shape, the roots, the turning point and the y-intercept clearly labelled.

Practice Questions

Question 1 (Foundation): Plot y = x² − 2x − 3 for x = −2 to 4 and state the roots of x² − 2x − 3 = 0.

Answer: The table gives points (−2, 5), (−1, 0), (0, −3), (1, −4), (2, −3), (3, 0), (4, 5). Roots are x = −1 and x = 3.

Question 2 (Higher): By completing the square, find the turning point of y = x² + 6x + 5.

Answer: y = (x + 3)² − 9 + 5 = (x + 3)² − 4. Turning point is (−3, −4). It is a minimum.

Question 3 (Higher): Sketch the graph of y = −x² + 4x − 3. State the roots, y-intercept and turning point.

Answer: y = −(x² − 4x) − 3 = −[(x − 2)² − 4

− 3 = −(x − 2)² + 4 − 3 = −(x − 2)² + 1. Turning point: (2, 1) maximum. Roots: −x² + 4x − 3 = 0 → x² − 4x + 3 = 0 → (x − 1)(x − 3) = 0 → x = 1, x = 3. y-intercept: (0, −3).]

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Summary

A quadratic graph is a parabola with equation y = ax² + bx + c.
If a > 0, the parabola is U-shaped (minimum turning point); if a < 0, it is ∩-shaped (maximum).
Roots are where the curve crosses the x-axis; the y-intercept is the value of c.
The line of symmetry passes through the turning point at x = −b/(2a).
Completing the square gives the turning point form y = a(x − p)² + q, with turning point (p, q).
When plotting, always use a smooth curve — never straight-line segments.
At Higher tier, you must be able to sketch quadratics showing roots, turning point and y-intercept.

Quadratic Graphs