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Maths · Equations and graphs

Using quadratic graphs

Draw quadratic graphs from a table, find roots, the turning point and the line of symmetry, and use a graph to solve related equations.

  • 6 key terms
  • All boards
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Warm-up

Answer each one, then check.

  1. 1

    Work out \(x^2 - 2x - 3\) when \(x = 2\).

    Show answerHide answer

    \(-3\)

  2. 2

    What shape is a graph of \(y = x^2\)?

    Show answerHide answer

    A U-shaped parabola

  3. 3

    What is a root of a graph?

    Show answerHide answer

    Where it crosses the x-axis

  4. 4

    What is a line of symmetry?

    Show answerHide answer

    A line the graph reflects onto itself in

  5. 5

    What is \(y\) on the y-axis?

    Show answerHide answer

    The vertical coordinate

Learning Objectives

  1. 1Complete a table of values and draw a quadratic graph.
  2. 2Find roots, the y-intercept, the turning point and the line of symmetry.
  3. 3Solve \(ax^2 + bx + c = 0\) using a graph.
  4. 4Solve related equations such as \(x^2 - 2x - 3 = 2\) by drawing a line.

QUADRATIC GRAPH

A quadratic graph is a smooth curve called a parabola. It is symmetrical about a vertical line through its turning point.

The roots are where \(y = 0\). To solve \(f(x) = k\), draw the line \(y = k\) and read the \(x\)-values where it meets the curve.

Table of Values

\(y = x^2 - 2x - 3\).

  • \(y\)

    \(-2\): \(5\). \(-1\): \(0\). \(0\): \(-3\). \(1\): \(-4\) | \(-3\) | \(0\) | \(5\)

Roots and Turning Point

For \(y = x^2 - 2x - 3\), find the roots, the turning point and the line of symmetry.

Show the solutionHide the solution
  1. 1 Roots \(y = 0\): \((x - 3)(x + 1) = 0\), so \(x = 3\) or \(-1\)
  2. 2 Line of symmetry Halfway between the roots: \(x = 1\)
  3. 3 Turning point \(y = 1 - 2 - 3 = -4\), so \((1, -4)\)

AnswerRoots \(-1\) and \(3\); line of symmetry \(x = 1\); minimum point \((1, -4)\).

Solving a Related Equation

Use the graph of \(y = x^2 - 2x - 3\) to solve \(x^2 - 2x - 3 = 2\).

Show the solutionHide the solution
  1. 1 Draw the line \(y = 2\) across the graph
  2. 2 Read the \(x\)-values where it meets the curve \(x \approx -1.4\) and \(x \approx 3.4\)
  3. 3 Check \(1 \pm \sqrt{6} = -1.45,\ 3.45\)

Answer\(x \approx -1.4\) or \(x \approx 3.4\)

Completed Square Form

Write \(y = x^2 - 2x - 3\) in the form \((x - a)^2 + b\) and write down the turning point.

Show the solutionHide the solution
  1. 1 Complete the square \((x - 1)^2 - 1 - 3\)
  2. 2 Simplify \(y = (x - 1)^2 - 4\)
  3. 3 Turning point \((1, -4)\)

Answer\(y = (x - 1)^2 - 4\); the turning point is \((1, -4)\).

Drawing Tips

Get a smooth curve.

  • Plot every point

    Then join with a smooth curve, not straight segments.

  • No flat bottom

    Near the turning point the curve rounds gently.

  • Check symmetry

    Values on each side of the turning point should match.

  • Use a ruler for lines

    Use a pencil for the curve.

Sketch and Solve

Complete the table for \(y = x^2 - 4x\) for \(x = 0\) to \(4\) and sketch the graph. Write down the roots and the turning point.

1. Substitute each x value.

2. Join with a smooth curve.

A good answer shows: \(y = 0, -3, -4, -3, 0\). Roots \(x = 0\) and \(x = 4\); turning point \((2, -4)\).

Can I...?

  1. 1Complete a table.
  2. 2Plot the points.
  3. 3Draw a smooth curve.
  4. 4Find the roots.
  5. 5Find the turning point.
  6. 6Give the line of symmetry.
  7. 7Solve a related equation.
  8. 8Use the completed square form.

Summary & Exam Focus

  • Roots are where \(y = 0\).
  • The turning point lies on the line of symmetry.
  • Solve \(f(x) = k\) with the line \(y = k\).
  • Completed square form gives the turning point directly.

Exam focus

The graph of \(y = x^2 - 2x - 3\) is drawn. Use it to solve \(x^2 - 2x - 3 = 0\) and to find the coordinates of the turning point. (3 marks) (3 marks)

Draw your line across the graph and mark the intersections.

Key terms

The vocabulary this lesson expects you to use. Each one is linked from the first place it appears above.

Parabola
The U-shaped graph of a quadratic.
Root
A value of \(x\) where \(y = 0\).
Turning point
The lowest or highest point of the curve.
Line of symmetry
The vertical line through the turning point.
\(y\)-intercept
Where the graph crosses the y-axis.
Minimum
The lowest point of a U-shaped curve.

Practice questions

Have a go at each one before you open its answer.

  1. Question 1 Use the graph 3 marks

    The graph of \(y = x^2 - 2x - 3\) is drawn on the grid. (a) Use the graph to solve \(x^2 - 2x - 3 = 0\). (b) Write down the coordinates of the turning point.

    The graph of y equals x squared minus 2x minus 3 on a grid.
    Show answerHide answer

    Model answer

    (a) \(x = -1\) and \(x = 3\). (b) \((1, -4)\).

    Mark scheme

    • \(-1\) — B1
    • 3 — B1
    • \((1, -4)\) — B1
  2. Question 2 Use the graph 2 marks

    Use the same graph to find estimates for the solutions of \(x^2 - 2x - 3 = 2\).

    Show answerHide answer

    Model answer

    Draw \(y = 2\). The solutions are \(x \approx -1.4\) and \(x \approx 3.4\). Accept \(-1.6\) to \(-1.3\) and \(3.3\) to \(3.6\).

    Mark scheme

    • Draws \(y = 2\) — M1
    • Both estimates — A1
  3. Question 3 Complete the table 2 marks

    Complete the table of values for \(y = x^2 - 2x - 3\) for \(x = -2, -1, 0, 1, 2, 3, 4\).

    Show answerHide answer

    Model answer

    \(y = 5, 0, -3, -4, -3, 0, 5\)

    Mark scheme

    • At least 4 correct — M1
    • All correct — A1
  4. Question 4 Write down 2 marks

    \(y = (x - 1)^2 - 4\). Write down the coordinates of the turning point of the graph and the equation of its line of symmetry.

    Show answerHide answer

    Model answer

    Turning point \((1, -4)\); line of symmetry \(x = 1\).

    Mark scheme

    • \((1, -4)\) — B1
    • \(x = 1\) — B1
  5. Question 5 Sketch 3 marks

    Sketch the graph of \(y = x^2 - 4\), showing where it crosses the axes.

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    Model answer

    A U-shaped parabola with its minimum at \((0, -4)\), crossing the x-axis at \((-2, 0)\) and \((2, 0)\), and the y-axis at \((0, -4)\).

    Mark scheme

    • Correct U shape — B1
    • x-intercepts \(\pm 2\) — B1
    • y-intercept \(-4\) — B1
  6. Question 6 Explain 2 marks

    A quadratic graph has roots at \(x = 2\) and \(x = 6\). Write down the equation of its line of symmetry and explain how you know.

    Show answerHide answer

    Model answer

    \(x = 4\); it is halfway between the roots.

    Mark scheme

    • \(x = 4\) — B1
    • Halfway between the roots — C1

Quick check

  1. The roots of a graph are where...

    1. A\(x = 0\)
    2. BIt is steepest
    3. C\(y = 0\)
    4. DIt turns
    Show answerHide answer

    C: \(y = 0\)

    \(y = 0\).

  2. A U-shaped quadratic graph has a...

    1. AMinimum point
    2. BMaximum point
    3. CNo turning point
    4. DStraight side
    Show answerHide answer

    A: Minimum point

    Minimum turning point.

  3. Roots \(-1\) and \(3\) give a line of symmetry at...

    1. A\(x = 0\)
    2. B\(x = 1\)
    3. C\(x = 2\)
    4. D\(x = 3\)
    Show answerHide answer

    B: \(x = 1\)

    Halfway: \(x = 1\).

  4. To solve \(x^2 - 2x - 3 = 2\) using the graph, draw...

    1. A\(x = 2\)
    2. B\(y = 0\)
    3. C\(y = x\)
    4. D\(y = 2\)
    Show answerHide answer

    D: \(y = 2\)

    The line \(y = 2\).

  5. \(y = (x - 3)^2 + 5\) has turning point...

    1. A\((-3, 5)\)
    2. B\((3, -5)\)
    3. C\((3, 5)\)
    4. D\((5, 3)\)
    Show answerHide answer

    C: \((3, 5)\)

    \((3, 5)\).

  6. A quadratic with negative \(x^2\) coefficient looks...

    1. ALike a U
    2. BLike an upside-down U
    3. CLike a straight line
    4. DLike an S
    Show answerHide answer

    B: Like an upside-down U

    Like an upside-down U, with a maximum point.

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