A-level · Algebra & functions

Transformations of graphsExam-style questions

Moving, stretching and reflecting a graph by changing its equation. Why a change outside the bracket acts on y as written; why a change inside acts on x and works backwards; stretches parallel to the x-axis and their scale factor; the two reflections; tracking a point through a combined transformation; and writing the equation of a transformed curve.

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13 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
Describe a sequence of two geometrical transformations that maps the curve $y = \sqrt{x}$ onto the curve $y = 2\sqrt{x + 5}$ .

(3 marks)
Level 2 · 3 marksTry it
2
The curve with equation $y = \text{f}(x)$ , $x \in \mathbb{R}$ , passes through the point $P(5,\ -2)$

Find the coordinates of the point to which $P$ is mapped on each of the following curves.
(a) $y = \text{f}(x) - 5$

(1 mark)
(b) $y = |\text{f}(x)|$

(1 mark)
(c) $y = 3\text{f}(x + 1) - 4$

(2 marks)
Level 2 · 4 marksTry it
3
The point $R(-3,\ 4)$ lies on the curve with equation $y = \text{f}(x)$ , $x \in \mathbb{R}$

Find the coordinates of the point to which $R$ is mapped when the curve with equation $y = \text{f}(x)$ is transformed to the curve with equation
(a) $y = \text{f}(x) + 7$

(1 mark)
(b) $y = \text{f}(x - 6)$

(1 mark)
(c) $y = 4\text{f}(-x + 1) - 9$

(2 marks)
Level 2 · 4 marksTry it
4
The point $Q(-8,\ 6)$ is on the curve $y = \text{f}(x)$ , $x \in \mathbb{R}$

For each of the transformed curves below, find the coordinates of the image of $Q$ .
(a) $y = \text{f}(x + 4)$

(1 mark)
(b) $y = \text{f}(4x)$

(1 mark)
(c) $y = 5\text{f}(-x) - 3$

(2 marks)
Level 2 · 4 marksTry it
5
The curve $y = x^2 - 4x + 7$ is translated by the vector $\begin{pmatrix} 2 \\ -3 \end{pmatrix}$ to give the curve $C$ .
(a) Show that $C$ has equation $y = (x - 4)^2$ . (3 marks)
(b) Write down the coordinates of the vertex of $C$ . (1 mark)
(c) State, with a reason, the number of real roots of the equation $(x - 4)^2 = 0$ , and what this means geometrically for $C$ . (2 marks)
Level 3 · 6 marksTry it

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