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
The curve $y = f(x)$ passes through the origin and has a single maximum point at $(3, 4)$ . The function $g$ is defined by $g(x) = f(x + 1) - 2$ . State the coordinates of (a) the maximum point of $y = g(x)$ , and (b) the point on $y = g(x)$ corresponding to the origin on $y = f(x)$ .
Level 4 · 2 marksTry it
2
(a) The curve $y = 2x^{2} - 5x + 1$ is translated by $\begin{pmatrix} -3 \\ 4 \end{pmatrix}$ .

Find and simplify the equation of the translated curve.

(2 marks)
(b) The graph of $y = \mathrm{f}(x)$ is transformed to the graph of $y = -\mathrm{f}\left(\tfrac12 x\right)$ .

Describe fully the two single transformations which have been combined to give the resulting transformation.

(3 marks)
Level 4 · 5 marksTry it
3
(a) The curve $y = 3^x$ can be mapped onto the curve $y = 3^{x + 2}$ by a single translation. The same mapping can also be achieved by a single stretch.

(i) Give full details of the translation.

(ii) Give full details of the stretch.

(4 marks)
(b) In this question you must show detailed reasoning.

Determine the value of $x$ for which $2\log_3 x - \log_3(x + 6) = 1$ .

(4 marks)
Level 4 · 8 marksTry it
4
(a) The point $A(4, -2)$ lies on the curve $y = f(x)$ , where f is a one-one function. Each of the curves below is the image of $y = f(x)$ under one or more transformations. Write down the coordinates of the image of $A$ on each curve.

(i) $y = f\left(\tfrac12 x\right)$

(ii) $y = 3 - 2f(x)$

(iii) $y = f^{-1}(x)$

(4 marks)
(b) The diagram shows part of the curve $y = g'(x)$ , where $g'(x)$ is the derivative of a function $g(x)$ .The curve $y = g'(x)$ crosses the $x$ -axis at $x = -3$ , $x = 0$ and $x = 5$ only. Its turning points are where $x = -\tfrac53$ and $x = 3$ .

(i) Write down the $x$ -coordinates of the stationary points on the curve $y = g(x)$ .

(ii) Find the set of values of $x$ for which $g(x)$ is decreasing.

(iii) Write down the $x$ -coordinates of the points of inflection on the curve $y = g(x)$ .

(3 marks)
Level 4 · 7 marksTry it
5
Figure 1

Figure 1 shows a sketch of the curve with equation $y = \mathrm{f}(x)$ .

The curve has a minimum point at $(-2, 0)$ , crosses the $y$ -axis at $(0, 8)$ and crosses the $x$ -axis at $(4, 0)$ .

On separate diagrams, sketch the curve with equation
(a) $y = \mathrm{f}(x - 2)$

(3 marks)
(b) $y = \mathrm{f}(-2x)$

(3 marks)

On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.
Level 4 · 6 marksTry it

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