Combinations of Transformations (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

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

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

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

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

On separate diagrams, sketch the following graphs.

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

(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 $y=\text{f}\left(x\right)$ onto each of the following graphs.

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

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

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

1. a translation by $\left(\begin{array}{c}2\\ 0\end{array}\right)$

2. a vertical stretch with scale factor 4

3. a translation by $\left(\begin{array}{c}0\\ -3\end{array}\right)$

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

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

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

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

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

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

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

The diagram shows the graph of $y=\text{f}\left(x\right)$, where $\text{f}\left(x\right)=(x-a{)}^2$ 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 $y=\text{f}\left(t\right)$, where $\text{f}\left(t\right)=\text{sin} 2t$ for $0°\le t\le 180°$.

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

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

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

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

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

The graph of $y=\text{f}\left(x\right)$ 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 $\left(\begin{array}{c}0\\ 2\end{array}\right)$

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

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

1. a translation by $\left(\begin{array}{c}2\\ 0\end{array}\right)$

2. a reflection in the $y$-axis

3. a vertical stretch with scale factor $\frac{2}{3}$

4. a translation by $\left(\begin{array}{c}0\\ 4\end{array}\right)$

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

The diagram shows the graph of $y=\text{f}\left(x\right)$. 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) $y=\text{f}\left(\frac{1}{2}x\right)+2$

(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 $y=\text{f}\left(x\right)$ onto each of the following graphs.

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

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

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

1. a translation by $\left[\begin{array}{c}-3\\ 0\end{array}\right]$

2. a horizontal stretch with scale factor $\frac{1}{2}$

3. a reflection in the $x$-axis

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

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

Sketch the graph of $y=-3\text{f}\left(2x\right)$, 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 $\text{f}\left(x\right)$.

The diagram shows the graph of $y=\text{f}\left(t\right)$, where $\text{f}\left(t\right)=\text{cos} t$ for $0°\le t\le 360°$.

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

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

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

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

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

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

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

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

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

The diagram shows the graph of $y=\text{f}\left(x\right)$, where $a$ is a positive constant. The maximum point $C(0,-4a)$, 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 $y=\text{f}\left(x\right)$ 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 $\text{f}\left(x\right)=kx$, where $k$ is a non-zero constant.

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

The diagram shows the graph of $y=\text{f}\left(x\right)$, where $\text{f}\left(x\right)=(x^2-a{)}^2$ 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 $\text{f}\left(x\right)$, 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 $(x-3{)}^2-(y-1{)}^2=1$.