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Quadratic Graphs

FoundationHigherAQAEdexcelOCR

Get to grips with quadratic graphs using these Foundation and Higher GCSE Maths practice questions. The worksheet focuses on plotting and interpreting quadratic graphs, and the accompanying mark scheme breaks down each solution clearly. Suitable for AQA, Edexcel and OCR. Download the questions and answers for free. A positive x² gives a U-shape; a negative x² gives an n-shape.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

A quadratic graph is the curve produced by an equation containing an \(x^2\) term. The shape is called a parabola, and it is always symmetrical.

The sign of the \(x^2\) coefficient decides which way it opens. A positive coefficient gives a U shape with a minimum point, and a negative coefficient gives an upside-down curve with a maximum.

Plotting is done from a table of values, exactly as for a straight line, but the curve must be drawn smoothly rather than with a ruler. Joining the points with straight segments loses marks, because a quadratic never contains a straight section.

Revision notes

Building the table

Substitute each \(x\) value into the equation, taking care with negatives.

For \(y = x^2 - 3\) with \(x = -2, -1, 0, 1, 2\): the \(y\) values are \(1, -2, -3, -2, 1\). Notice the symmetry, which is a useful check that no value has been miscalculated.

Drawing the curve

Plot the points and join them with a single smooth curve, turning gently at the lowest or highest point.

The turning point often falls between two plotted values, so the curve should be drawn through the gap rather than pointed at a plotted point.

Reading the key features

The roots are where the curve crosses the \(x\)-axis, and they are the solutions of the equation when \(y = 0\).

The turning point is the minimum or maximum, and the curve is symmetrical about a vertical line through it.

Key points

  • A quadratic graph is a parabola.
  • A positive \(x^2\) coefficient gives a U shape.
  • A negative \(x^2\) coefficient gives an upside-down curve.
  • The curve is symmetrical.
  • Draw a smooth curve, never straight segments.
  • Roots are where the curve crosses the x-axis.

Worked examples

Example 1

Complete the table for \(y = x^2 + 1\) at \(x = -2, 0, 2\).

Working

\[(-2)^2 + 1 = 5\]substitute x = −2, squaring the negative
\[0^2 + 1 = 1\]substitute x = 0
\[2^2 + 1 = 5\]substitute x = 2, matching the value at −2

Example 2

Find \(y\) when \(x = -3\) for \(y = x^2 - 2x\).

Working

\[(-3)^2 = 9\]square the negative first
\[-2 \times -3 = 6\]multiply, noting two negatives give a positive
\[9 + 6 = 15\]add to find y

Example 3

State the shape of the graph \(y = -x^2 + 4\).

Working

\[\text{Coefficient of } x^2 \text{ is } -1\]identify the sign
\[\text{An upside-down U with a maximum}\]a negative coefficient opens downwards

Common mistakes

  • Joining the points with straight lines.

    A quadratic is a smooth curve throughout, with no straight sections.

  • Sign errors when squaring negatives.

    (−3)² is 9, not −9. Square the whole value including its sign.

  • Making the turning point a sharp corner.

    The curve turns gently, and often between two plotted points.

  • Not using the symmetry as a check.

    Matching y values either side of the turning point confirm the table is right.

Exam tips

  • Use the symmetry of the table to check your values.
  • Draw the curve freehand and smoothly, not with a ruler.
  • Plot enough points either side of the turning point.
  • Check the direction of the curve against the sign of the x² term.

Key terms

Parabola
The symmetrical curve produced by a quadratic equation.
Root
A value of x where the curve crosses the x-axis.
Turning point
The maximum or minimum point of the curve.
Coefficient
The number multiplying a term.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.