OpenRevise GCSE

Viewing as

Student Teacher

Teacher view: planning notes, the answers to every question, and the teacher copies of the files.

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
Download the teacher pack · 5 files

Teacher resources

The teacher copies: slides with the questions built in, the answers, and anything else attached to this lesson for whoever is teaching it.

Student handouts

The same files the students see, to print or hand out.

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.

Questions and answers

12 questions set on this lesson, with the mark schemes and model answers open.

1. Exam question Use the graph 3 marks Easier

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.

Mark scheme — 3 marks available

  • \(-1\) — B1
  • 3 — B1
  • \((1, -4)\) — B1

Model answer

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

2. Exam question Use the graph 2 marks Easier

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

Mark scheme — 2 marks available

  • Draws \(y = 2\) — M1
  • Both estimates — A1

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 \(3.6\).

3. Exam question Complete the table 2 marks Easier

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

Mark scheme — 2 marks available

  • At least 4 correct — M1
  • All correct — A1

Model answer

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

4. Exam question Write down 2 marks Easier

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

Mark scheme — 2 marks available

  • \((1, -4)\) — B1
  • \(x = 1\) — B1

Model answer

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

5. Exam question Sketch 3 marks Easier

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

Mark scheme — 3 marks available

  • Correct U shape — B1
  • x-intercepts \(\pm 2\) — B1
  • y-intercept \(-4\) — B1

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 \((0, -4)\).

6. Exam question Explain 2 marks Easier

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.

Mark scheme — 2 marks available

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

Model answer

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

7. Multiple choice 1 mark Easier

The roots of a graph are where...

  1. A \(x = 0\)
  2. B It is steepest
  3. C \(y = 0\) Correct
  4. D It turns

Why: \(y = 0\).

8. Multiple choice 1 mark Core

A U-shaped quadratic graph has a...

  1. A Minimum point Correct
  2. B Maximum point
  3. C No turning point
  4. D Straight side

Why: Minimum turning point.

9. Multiple choice 1 mark Core

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

  1. A \(x = 0\)
  2. B \(x = 1\) Correct
  3. C \(x = 2\)
  4. D \(x = 3\)

Why: Halfway: \(x = 1\).

10. Multiple choice 1 mark Core

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 \(y = 2\) Correct

Why: The line \(y = 2\).

11. Multiple choice 1 mark Core

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

  1. A \((-3, 5)\)
  2. B \((3, -5)\)
  3. C \((3, 5)\) Correct
  4. D \((5, 3)\)

Why: \((3, 5)\).

12. Multiple choice 1 mark Stretch

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

  1. A Like a U
  2. B Like an upside-down U Correct
  3. C Like a straight line
  4. D Like an S

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