Sheet № 26 · Foundation + Higher · AQA · Edexcel · OCR
Quadratic Graphs
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
§Key definitions
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.
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).
Step 1:
Factor out the coefficient of x² from the first two terms:
Step 2:
Complete the square inside the bracket. Half of −6 is −3, and (−3)² = 9:
§Formulas to memorise
y = ax² + bx + c
x = −b / (2a)
y = a(x − p)² + q
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:
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.
Worked example
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:
⚠ 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.