Combinations of Transformations (Cambridge International AS Mathematics): Exam Questions

The diagram below shows the graph of $y = \text{f} ( x )$.

The stationary point $A ( - 2 , 8 )$ is marked on the diagram.

On separate diagrams, sketch the following graphs.

(i) $y = 2 \text{f} ( x ) + 1$

(ii) $y = \frac{1}{2} \text{f} ( x + 1 )$

On each diagram, state the coordinates of the image of point $A$ under the given transformation.

The graph of $y = \text{f} ( x )$ is transformed. In each case, write down, in terms of $\text{f} ( x )$, the equation of the resulting graph.

(i) a translation by \begin{pmatrix} -2 \\ 3 \end{pmatrix}

(ii) a horizontal stretch with scale factor 2, followed by a vertical stretch with scale factor 3

The function $\text{g}$ is defined by $\text{g} ( x ) = 2 x$.

On the same diagram, sketch the graphs of $y = \text{g} ( x )$ and $y = \text{g}^{-1} ( x )$, and indicate the line of symmetry.

Label the coordinates of any point where the graphs cross the coordinate axes.

(i) Write down an expression for $\text{g}^{-1} ( x )$ in terms of $x$.

(ii) Find an expression for $\text{g}^{-1} ( x )$ in terms of $\text{g} ( x )$, and state the single transformation this represents.

The diagram shows the graph of $y = \text{f} ( x )$, where $\text{f} ( x ) = ( x - 2 )^{2}$.

The graph meets the $y$-axis at $A$ and touches the $x$-axis at $B$.

(i) Write down the coordinates of $A$ and $B$.

(ii) The graph of $y = \text{f} ( x )$ is transformed to the graph of $y = - \text{f} ( x ) - 4$. Find the coordinates of the images of $A$ and $B$ under this transformation.

The diagram shows the graph of $y = \text{f} ( t )$, where $\text{f} ( t ) = \text{cos} t$ for $0 ^{\circ} \leq t \leq 360 ^{\circ}$.

(i) Write down the minimum value of $\text{f} ( t )$ in the given domain.

(ii) Write down the value of $t$ at which this minimum occurs.

(i) Write down the minimum value of $y = 3 \text{f} ( t - 45 ^{\circ} )$ in the given domain.

(ii) Write down the value of $t$ at which this minimum occurs.

Describe the combination of transformations that maps the graph of $y = \text{f} ( t )$ onto the graph of $y = 3 \text{cos} t + 1$ for $0 ^{\circ} \leq t \leq 360 ^{\circ}$.

The graph of $y = \text{f} ( x )$ is transformed by the following sequence of transformations, applied in this order.

1. a reflection in the $y$-axis
2. a vertical stretch with scale factor 4
3. a translation by \begin{pmatrix} 0 \\ -2 \end{pmatrix}

Write down, in terms of $\text{f} ( x )$, the equation of the resulting graph.

The diagram shows the graph of $y = \text{g} ( x )$, where $\text{g} ( x ) = \frac{1}{x}$, $x \neq 0$.

(i) Write down the equations of the two asymptotes of $y = \text{g} ( x )$.

(ii) Determine the equations of the two asymptotes of the graph of $y = \text{g} ( x - 1 ) + 5$.

Describe fully, in order, the sequence of transformations that maps the graph of $y = \text{f} ( x )$ onto each of the following graphs.

(i) $y = 3 \text{f} ( x ) - 1$

(ii) $y = \frac{1}{3} \text{f} ( x - 1 )$

The diagram shows the graph of $y = \text{f} ( x )$. The stationary points $A ( - 3 , 27 )$ and $B ( 1 , - 5 )$ are marked on the diagram.

On separate diagrams, sketch the following graphs.

(i) $y = 2 \text{f} ( x ) - 4$

(ii) $y = \text{f} ( x + 1 ) + 3$

On each diagram, state the coordinates of the images of $A$ and $B$ under the given transformation.

Describe fully, in order, the sequence of transformations that maps the graph of $y = \text{f} ( x )$ onto each of the following graphs.

(i) $y = 3 \text{f} ( x + 2 )$

(ii) $y = \text{f} ( - x ) - 1$

Given that $\text{f} ( x ) = 3 x^{2} - 2 x$, find an expression for $\text{g} ( x )$, where $\text{g} ( x )$ is obtained by applying the following sequence of transformations to $y = \text{f} ( x )$, in this order.

1. a translation by \begin{pmatrix} 2 \\ 0 \end{pmatrix}
2. a vertical stretch with scale factor 4
3. a translation by \begin{pmatrix} 0 \\ -3 \end{pmatrix}

The function $\text{p}$ is defined by $\text{p} ( x ) = 3 x - 4$.

(i) Sketch the graph of $y = \text{p} ( x )$.

(ii) On the same axes, sketch the graph of $y = \text{p}^{-1} ( x )$, indicating the line of symmetry.

Label the coordinates of the points where each graph crosses the coordinate axes.

(i) Find an expression for $\text{p}^{-1} ( x )$.

(ii) Find an expression for \tfrac{1}{9}\left[\text{p}(x) + 16\right], giving your answer in its simplest form.

(iii) Hence state what the sequence of transformations represented by \tfrac{1}{9}\left[\text{p}(x) + 16\right] does to the graph of $y = \text{p} ( x )$.

The diagram shows the graph of $y = \text{f} ( x )$, where $\text{f} ( x ) = ( x - a )^{2}$ and $a > 1$.

The points $A$ and $B$ are where the graph meets the coordinate axes.

Write down, in terms of $a$, the coordinates of $A$ and $B$.

Sketch the graph of $y = - \text{f} ( - x )$, labelling the images of $A$ and $B$ and stating their coordinates in terms of $a$.

Given that $A$ is three times as far from the origin as $B$, find the value of $a$.

The diagram shows the graph of $y = \text{f} ( t )$, where $\text{f} ( t ) = \text{sin} 2 t$ for $0 ^{\circ} \leq t \leq 180 ^{\circ}$.

(i) Write down the maximum value of $y = 3 \text{f} ( t )$.

(ii) Write down the first value of $t$ at which this maximum occurs.

(i) Write down the minimum value of $y = 5 \text{f} ( t + 30 ^{\circ} )$.

(ii) Write down the first value of $t$ at which this minimum occurs.

Describe, in terms of $\text{f} ( t )$, the combination of transformations that maps the graph of $y = \text{f} ( t )$ onto the graph of $y = 2 + \text{sin} t$ for $0 ^{\circ} \leq t \leq 180 ^{\circ}$.

The graph of $y = \text{f} ( x )$ is transformed by the following sequence of transformations, applied in this order.

1. a horizontal stretch with scale factor 2
2. a reflection in the $x$-axis
3. a translation by \begin{pmatrix} 0 \\ 2 \end{pmatrix}

Write down, in terms of $\text{f} ( x )$, the equation of the resulting graph.

The graph of $y = \text{f} ( x )$ is transformed by the following sequence of transformations, applied in this order.

1. a translation by \begin{pmatrix} 2 \\ 0 \end{pmatrix}
2. a reflection in the $y$-axis
3. a vertical stretch with scale factor $\frac{2}{3}$
4. a translation by \begin{pmatrix} 0 \\ 4 \end{pmatrix}

Write down, in terms of $\text{f} ( x )$, the equation of the resulting graph.

The diagram shows the graph of $y = \text{f} ( t )$, where $\text{f} ( t ) = \text{cos} t$ for $0 ^{\circ} \leq t \leq 360 ^{\circ}$.

(i) Write down the maximum value of $y = - 2 \text{f} ( 3 t )$.

(ii) Write down the value of $t$ at which this maximum first occurs.

Find, in terms of $\text{f} ( t )$, a combination of transformations that maps the graph of $y = \text{f} ( t )$ onto the graph of $y = 2 - 4 \text{sin} t$ for $0 ^{\circ} \leq t \leq 360 ^{\circ}$.

The diagram shows the graph of $y = \text{g} ( x )$, where $\text{g} ( x ) = \frac{1}{ax}$, with $a$ a non-zero constant.

(i) Write down the equations of the asymptotes of $y = \text{g} ( x )$.

(ii) Determine the equations of the asymptotes of the graph of $y = 3 \text{g} ( 2 x + 1 )$.

The function $\text{f}$ is defined by \text{f}(x) = -2\text{g}\left(\tfrac{1}{3}x + 3\right) - 4. Determine the domain and range of $\text{f}$.

The function $\text{p}$ is defined by $\text{p} ( x ) = 2 x + 1$.

Show that the graph of $y = \text{p} ( x )$ maps onto the graph of its inverse under the transformation $\frac{1}{2} \text{p} ( \frac{1}{2} x ) - 1$.

The diagram shows the graph of $y = \text{f} ( x )$. The maximum point $B ( - 1 , 32 )$ and the $x$-axis intercepts $A ( - 3 , 0 )$ and $C ( 3 , 0 )$ are marked on the diagram.

On separate diagrams, sketch the following graphs.

(i) $y = \text{f} ( \frac{1}{2} x ) + 2$

(ii) $y = - \text{f} ( x - 1 )$

On each diagram, mark the coordinates of the images of the points $A$, $B$ and $C$ under the given transformation.

Describe fully, in order, the sequence of transformations that maps the graph of $y = \text{f} ( x )$ onto each of the following graphs.

(i) $y = - \text{f} ( 3 x - 1 )$

(ii) $y = 2 \text{f} ( 5 - x )$

The function $\text{f}$ is defined by $\text{f} ( x ) = \sqrt{2x+1}$. Find an expression for $\text{g} ( x )$, where $\text{g} ( x )$ is obtained by applying the following sequence of transformations to $y = \text{f} ( x )$, in this order.

1. a translation by \begin{bmatrix} - 3 \\ 0 \end{bmatrix}
2. a horizontal stretch with scale factor $\frac{1}{2}$
3. a reflection in the $x$-axis

The diagram shows the graph of $y = \text{f} ( x )$, where $\text{f} ( x ) = ( x^{2} - 4 )^{2}$.

The points $A ( - 2 , 0 )$, $B ( 0 , 16 )$ and $C ( 2 , 0 )$ are where the graph meets the coordinate axes.

Sketch the graph of $y = - 3 \text{f} ( 2 x )$, labelling the images of $A$, $B$ and $C$.

Suggest a combination of at least two transformations that maps $A$, $B$ and $C$ so that none of them lies on a coordinate axis. Give your answer as an expression in terms of $\text{f} ( x )$.

The diagram shows the graph of $y = \text{f} ( x )$, where $a$ is a positive constant. The maximum point $C ( 0 , - 4 a )$, the minimum point $B ( 2 - a , - 20 )$ and the $x$-axis intercept $A ( - a , 0 )$ are marked on the diagram.

On separate diagrams, sketch the following graphs, marking the coordinates of the images of $A$, $B$ and $C$ in terms of $a$.

(i) $y = 2 \text{f} ( \frac{1}{3} x - 1 )$

(ii) $y = - 2 \text{f} ( x + 1 ) + 1$

Describe, in order, a sequence of transformations that maps the graph of $y = \text{f} ( x )$ onto each of the following graphs.

(i) $y = a \text{f} ( x + b ) + c$, for the case when $a > 0$

(ii) $y = - \text{f} ( - x )$

How, if at all, would your answer to part (a)(i) change if $a = 1$, or if $a < 0$?

A function $\text{f}$ is defined by $\text{f} ( x ) = k x$, where $k$ is a non-zero constant.

Show that \text{f}^{-1}(x) = \tfrac{1}{k}\,\text{f}\left(\tfrac{1}{k}x\right).

The diagram shows the graph of $y = \text{f} ( x )$, where $\text{f} ( x ) = ( x^{2} - a )^{2}$ and $a > 1$. The points $A$, $B$ and $C$ are where the graph meets the coordinate axes.

Write down, in terms of $a$, the coordinates of $A$, $B$ and $C$.

Sketch the graph of $y = - \frac{1}{2} \text{f} ( x - 1 )$, labelling the images of $A$, $B$ and $C$ and stating their coordinates in terms of $a$.

Suggest, in terms of $\text{f} ( x )$, a combination of at least two transformations for which $A$, $B$ and $C$ move to new positions but each remains on its original axis.

The diagram shows the graph of $x^{2} - y^{2} = 1$, which has asymptotes $y = x$ and $y = - x$ and meets the $x$-axis at the points $A$ and $B$.

Determine the coordinates of the images of $A$ and $B$ on the graph of $( x - 3 )^{2} - ( y - 1 )^{2} = 1$.

(i) Determine the equations of the asymptotes of the graph of $( x - 3 )^{2} - ( y - 1 )^{2} = 1$.

(ii) Find the domain of $( x - 3 )^{2} - ( y - 1 )^{2} = 1$.