GCSE · Graphs of functions

Transformations of graphs (y = f(x) + a, y = f(x + a), y = −f(x), y = f(−x))Exam-style questions

Move a graph without plotting it again. A number added outside the f moves the curve up or down; a number added inside the bracket moves it sideways, the opposite way to its sign; a minus sign outside reflects it in the x-axis and a minus sign inside reflects it in the y-axis. Track the key points, describe each change fully, combine two changes, write the new equation, and use the same four rules on the sine and cosine waves.

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20 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
The graph of $y = \mathrm{h}(x)$ for $-3 \le x \le 2$ is made of three straight lines. It is drawn on the grid below.(a) On the grid, draw the graph of $y = -\mathrm{h}(x) + 2$

(2 marks)
(b) Write down the coordinates of the point where the graph of $y = -\mathrm{h}(x) + 2$ crosses the $y$ -axis.

(1 mark)
Level 4 · 3 marksTry it
2
The graph of $y = x^2 - 4x$ has a minimum point at $(2, -4)$
(a) The graph of $y = x^2 - 4x$ is reflected in the $y$ -axis.

Work out the equation of the reflected graph.

(1 mark)
(b) Which single transformation maps the graph of $y = x^2 - 4x$ onto the graph of $y = 4x - x^2$ ?

Tick one box.

☐ Reflection in the $x$ -axis

☐ Reflection in the $y$ -axis

☐ Rotation of $180°$ about the origin

☐ Translation by the vector $\begin{pmatrix} 0 \\ 4 \end{pmatrix}$

(1 mark)
(c) Write down the coordinates of the maximum point of the graph of $y = 4x - x^2$

(1 mark)
(d) The graph of $y = x^2 - 4x$ is translated by the vector $\begin{pmatrix} 0 \\ 5 \end{pmatrix}$

Does the translated graph cross the $x$ -axis?

Tick a box.

☐ Yes $\qquad$ ☐ No

Show working to support your answer.

(2 marks)
Level 4 · 5 marksTry it
3
(a) Show that $x^2 + 6x + 14$ can be written as $(x + 3)^2 + 5$

(2 marks)
(b) Describe fully the single transformation which maps the graph of $y = x^2$ onto the graph of $y = x^2 + 6x + 14$

(2 marks)
(c) The graph of $y = x^2 + 6x + 14$ is reflected in the $x$ -axis.

Write down the coordinates of the turning point of the reflected graph.

State whether it is a maximum or a minimum.

(2 marks)
Level 4 · 6 marksTry it
4
The sketch shows the graph of $y = \sin x$ for $0° \le x \le 360°$ (a) Write down the coordinates of the maximum point of the graph of $y = \sin x + 2$ for $0° \le x \le 360°$

(1 mark)
(b) The graph of $y = \sin x$ is reflected in the $x$ -axis.

Write down the equation of the reflected graph.

(1 mark)
(c) Write down the coordinates of the maximum point of the graph in part (b) for $0° \le x \le 360°$

(1 mark)
(d) The graph of $y = \sin x$ is translated by the vector $\begin{pmatrix} 60 \\ 0 \end{pmatrix}$

Find the values of $x$ , for $0° \le x \le 360°$ , where the translated graph crosses the $x$ -axis.

(2 marks)
Level 4 · 5 marksTry it
5
The grid shows the graph of $y = \mathrm{f}(x)$ and a curve $G$ . $G$ is the image of the graph of $y = \mathrm{f}(x)$ after a translation.
(a) Write down the equation of $G$ in terms of $\mathrm{f}$ .

(2 marks)
(b) The graph of $y = \mathrm{f}(x)$ is reflected in the $y$ -axis.

Write down the coordinates of the minimum point of the reflected graph.

(1 mark)
(c) The point $P(3, 2)$ lies on the graph of $y = \mathrm{f}(x)$

Write down the coordinates of the point that $P$ moves to on the graph of $y = 2 - \mathrm{f}(x)$

(1 mark)
Level 4 · 4 marksTry it

1 more question on transformations of graphs (y = f(x) + a, y = f(x + a), y = −f(x), y = f(−x))

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