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Maths · Functions, Sequences and Rates of Change

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Transformations of Graphs

Sketching and describing translations and reflections of graphs, including trigonometric graphs.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1Sketch and describe the graphs of \(y = f(x) + a\) and \(y = f(x + a)\), which are translations.
  2. 2Sketch and describe the graphs of \(y = -f(x)\) and \(y = f(-x)\), which are reflections.
  3. 3Say where a turning point or intercept moves to under a transformation.
  4. 4Write down the equation of a transformed graph, including trigonometric graphs.

Transforming a graph

If you know what the graph of \(y = f(x)\) looks like, you can sketch many related graphs without plotting points. Adding to the function moves the graph up or down. Changing the input moves the graph left or right, or reflects it. The key is to follow one important point, such as a turning point, and see where it goes. These are Higher tier skills on every board, and they apply to any graph, including the trigonometric graphs.

Translations

There are two kinds of translation, and the horizontal one often causes errors.

  • \(y = f(x) + a\)

    The graph moves up by \(a\), which is the translation \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).

  • \(y = f(x + a)\)

    The graph moves to the left by \(a\), which is the translation \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).

  • Opposite direction

    For \(f(x + a)\), the sign inside the bracket is the opposite of the direction of movement.

  • Check

    For \(y = (x - 3)^2\), the turning point is at \(x = 3\), so the graph of \(y = x^2\) has moved right by 3.

Moving a turning point

The graph of \(y = f(x)\) has a maximum point at \((3, 5)\). Write down the coordinates of the maximum point of \(y = f(x - 2)\) and of \(y = f(x) - 4\).

Show the solutionHide the solution
  1. 1 \(f(x - 2)\) The graph moves right by 2, so \(x\) becomes \(3 + 2 = 5\).
  2. 2 Maximum The \(y\)-coordinate stays 5, so the maximum is \((5, 5)\).
  3. 3 \(f(x) - 4\) The graph moves down by 4, so \(y\) becomes \(5 - 4 = 1\).
  4. 4 Maximum The \(x\)-coordinate stays 3, so the maximum is \((3, 1)\).

Answer\((5, 5)\) for \(y = f(x - 2)\), and \((3, 1)\) for \(y = f(x) - 4\)

Reflections

A minus sign reflects the graph in one of the axes.

  • \(y = -f(x)\)

    The minus is outside the function, so the graph is reflected in the \(x\)-axis. Every \(y\)-coordinate changes sign.

  • \(y = f(-x)\)

    The minus is inside the brackets, so the graph is reflected in the \(y\)-axis. Every \(x\)-coordinate changes sign.

  • A point \((a, b)\)

    Moves to \((a, -b)\) for \(-f(x)\), and to \((-a, b)\) for \(f(-x)\).

  • Memory aid

    Outside the bracket changes \(y\), and inside the bracket changes \(x\).

The equation of a transformed graph

The graph of \(y = x^2\) is translated 3 units to the right and then reflected in the \(x\)-axis. Write down the equation of the new graph.

Show the solutionHide the solution
  1. 1 Translate right Replace \(x\) by \(x - 3\) to get \(y = (x - 3)^2\).
  2. 2 Reflect in the x-axis Change the sign of the whole function.
  3. 3 New equation \(y = -(x - 3)^2\).
  4. 4 Check The turning point is \((3, 0)\), and the curve opens downwards, so it is a maximum.

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

Test yourself

  1. 1

    What does \(y = f(x) + a\) do?

    Show answerHide answer

    Moves the graph up by \(a\).

  2. 2

    What does \(y = f(x + a)\) do?

    Show answerHide answer

    Moves the graph left by \(a\).

  3. 3

    What does \(y = -f(x)\) do?

    Show answerHide answer

    Reflects the graph in the \(x\)-axis.

  4. 4

    What does \(y = f(-x)\) do?

    Show answerHide answer

    Reflects the graph in the \(y\)-axis.

  5. 5

    Where does \((4, 3)\) go under \(y = -f(x)\)?

    Show answerHide answer

    \((4, -3)\).

Exam technique: transformations

Track one point, and check with the equation.

  • Follow a point

    Move the turning point or an intercept, and redraw around it.

  • Label

    Write the coordinates of the new key points on the sketch.

  • Direction

    Remember that \(f(x + a)\) moves left, which is the opposite of what you might expect.

  • Describe fully

    For a description, say the type of transformation and give the vector or the axis.

Summary and exam focus

  • \(f(x) + a\) moves the graph up by \(a\), and \(f(x + a)\) moves it left by \(a\).
  • \(-f(x)\) reflects in the \(x\)-axis, and \(f(-x)\) reflects in the \(y\)-axis.
  • Follow a turning point to see where a graph goes.
  • The same rules apply to every graph, including sine and cosine.

Exam focus

The graph of \(y = f(x)\) has a minimum point at \((2, -1)\). Write down the coordinates of the minimum point of \(y = f(x + 1)\). (1 mark) (1 marks)

\(f(x + 1)\) moves the graph 1 unit to the left, so the minimum is at \((1, -1)\). The \(y\)-coordinate does not change.

Key terms

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

Transformation
A change to the position or shape of a graph.
Translation
A movement of a graph without turning or flipping it.
Reflection
A flip of a graph in a line, such as an axis.
Turning point
A point where a graph changes from rising to falling, or the reverse.
Intercept
A point where a graph crosses an axis.
Function notation
Writing a function as \(f(x)\).
Vector
A pair of numbers that gives a movement.
Maximum
The highest point of a graph.
Minimum
The lowest point of a graph.

Questions and answers

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

1. Exam question Write down 2 marks Core

The graph of \(y = f(x)\) is shown. The turning point is \((2, -1)\). (a) Write down the coordinates of the turning point of the graph of \(y = f(x) + 2\). (1 mark) (b) Write down the coordinates of the turning point of the graph of \(y = f(x + 1)\). (1 mark)

The graph of y equals f of x with its turning point marked.

Mark scheme — 2 marks available

  • (a) \((2, 1)\) — B1
  • (b) \((1, -1)\) — B1

Model answer

(a) \((2, 1)\). (b) \((1, -1)\).

2. Exam question Write down 2 marks Core

The graph of \(y = g(x)\) has a minimum point at \((-1, -3)\). (a) Write down the coordinates of the minimum point of \(y = g(x) + 4\). (1 mark) (b) Write down the coordinates of the maximum point of \(y = -g(x)\). (1 mark)

Mark scheme — 2 marks available

  • (a) \((-1, 1)\) — B1
  • (b) \((-1, 3)\) — B1

Model answer

(a) \((-1, 1)\). (b) \((-1, 3)\).

3. Exam question Write down 3 marks Core

The graph of \(y = x^2\) is transformed. Write down the equation of the new graph after (a) a translation of 3 units to the right, (1 mark) (b) a translation of 2 units up, (1 mark) (c) a reflection in the \(x\)-axis. (1 mark)

Mark scheme — 3 marks available

  • (a) \(y = (x - 3)^2\) — B1
  • (b) \(y = x^2 + 2\) — B1
  • (c) \(y = -x^2\) — B1

Model answer

(a) \(y = (x - 3)^2\). (b) \(y = x^2 + 2\). (c) \(y = -x^2\).

4. Exam question Write down 4 marks Core

The graph of \(y = f(x)\) has a maximum point at \((3, 5)\). Write down the coordinates of the maximum point of the graph of (a) \(y = f(x - 2)\) (1 mark) (b) \(y = f(x) - 4\) (1 mark) (c) \(y = -f(x)\) (1 mark) (d) \(y = f(-x)\) (1 mark)

Mark scheme — 4 marks available

  • (a) \((5, 5)\) — B1
  • (b) \((3, 1)\) — B1
  • (c) \((3, -5)\) — B1
  • (d) \((-3, 5)\) — B1

Model answer

(a) \((5, 5)\). (b) \((3, 1)\). (c) \((3, -5)\), which is now a minimum. (d) \((-3, 5)\).

5. Exam question Write down 3 marks Stretch

The diagram shows the graph of \(y = \sin x\) and a transformation of it, for \(0^\circ \le x \le 360^\circ\). (a) Write down the equation of the transformed graph. (1 mark) (b) Describe fully the single transformation. (2 marks)

The graph of y equals sin x and a transformed curve, a sine curve moved up by 1.

Mark scheme — 3 marks available

  • (a) \(y = \sin x + 1\) — B1
  • (b) Translation — B1
  • (b) Vector \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\), or 1 unit up — B1

Model answer

(a) \(y = \sin x + 1\). (b) A translation by the vector \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\), which is 1 unit up.

6. Exam question Show that 4 marks Stretch

\(f(x) = x^2 - 4x + 3\). Show that \(f(x + 1) = x^2 - 2x\), and hence solve \(f(x + 1) = 0\). (4 marks)

Mark scheme — 4 marks available

  • \((x + 1)^2 - 4(x + 1) + 3\) — M1
  • \(x^2 - 2x\) with the working shown — A1
  • \(x(x - 2) = 0\) — M1
  • \(x = 0\) and \(x = 2\) — A1

Model answer

\(f(x + 1) = (x + 1)^2 - 4(x + 1) + 3 = x^2 + 2x + 1 - 4x - 4 + 3 = x^2 - 2x\). Then \(x(x - 2) = 0\), so \(x = 0\) or \(x = 2\).

7. Multiple choice 1 mark Easier

What does \(y = f(x) + 3\) do to the graph of \(y = f(x)\)?

  1. A Moves it right 3
  2. B Moves it up 3 Correct
  3. C Moves it left 3
  4. D Moves it down 3

Why: Adding to the function moves the graph up.

8. Multiple choice 1 mark Easier

What does \(y = f(x + 2)\) do to the graph of \(y = f(x)\)?

  1. A Moves it left 2 Correct
  2. B Moves it right 2
  3. C Moves it up 2
  4. D Moves it down 2

Why: A plus inside the bracket moves the graph left.

9. Multiple choice 1 mark Easier

What does \(y = -f(x)\) do to the graph of \(y = f(x)\)?

  1. A Reflects it in the \(y\)-axis
  2. B Moves it down
  3. C Turns it by \(180^\circ\) about the origin
  4. D Reflects it in the \(x\)-axis Correct

Why: A minus outside changes the \(y\)-values.

10. Multiple choice 1 mark Core

What does \(y = f(-x)\) do to the graph of \(y = f(x)\)?

  1. A Reflects it in the \(x\)-axis
  2. B Moves it left
  3. C Reflects it in the \(y\)-axis Correct
  4. D Moves it down

Why: A minus inside changes the \(x\)-values.

11. Multiple choice 1 mark Core

The maximum of \(y = f(x)\) is at \((3, 5)\). Where is the maximum of \(y = f(x - 2)\)?

  1. A \((1, 5)\)
  2. B \((5, 5)\) Correct
  3. C \((3, 7)\)
  4. D \((3, 3)\)

Why: The graph moves right 2.

12. Multiple choice 1 mark Core

The maximum of \(y = f(x)\) is at \((3, 5)\). Where is the turning point of \(y = -f(x)\)?

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

Why: The \(y\)-coordinate changes sign.

13. Multiple choice 1 mark Core

What is the equation of \(y = x^2\) after a translation of 3 units to the right?

  1. A \(y = (x + 3)^2\)
  2. B \(y = x^2 + 3\)
  3. C \(y = x^2 - 3\)
  4. D \(y = (x - 3)^2\) Correct

Why: Moving right replaces \(x\) with \(x - 3\).

14. Multiple choice 1 mark Stretch

What is the maximum value of \(y = \sin x + 1\)?

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

Why: The sine graph is moved up by 1, so its maximum is \(1 + 1 = 2\).

15. Multiple choice 1 mark Stretch

Which equation gives the same graph as \(y = \cos x\)?

  1. A \(y = \sin(x - 90^\circ)\)
  2. B \(y = \sin(x + 90^\circ)\) Correct
  3. C \(y = -\sin x\)
  4. D \(y = \sin x + 90\)

Why: Moving the sine graph left by \(90^\circ\) gives the cosine graph.