Combinations of Transformations (OCR A Level Maths A: Pure): Exam Questions

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

The stationary points are marked on the diagram.

On separate diagrams, sketch the graphs with equation

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

(ii) $y=\mathrm{f}\left(x+1\right)+3$

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

Describe, in order, a sequence of transformations that maps the graph of $y=\mathrm{f}\left(x\right)$onto the following graphs:

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

(ii) $y=\mathrm{f}(—x)—1.$

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

(i) Translation by $\left(\begin{array}{c}2\\ 0\end{array}\right)$

(ii) Vertical stretch of scale factor $4$

(iii) Translation by $\left(\begin{array}{c}0\\ -3\end{array}\right)$

(i) Sketch the graph of $y=\mathrm{p}\left(x\right)$, where $\mathrm{p}\left(x\right)=3x-4.$

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

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

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

(ii) Find an expression for $\frac{1}{9}[\mathrm{p}\left(x\right)+16]$.

(iii) What can you deduce about the sequence of transformations given $\frac{1}{9}[p\left(x\right)+16]$?

The equation $y=f\left(x\right)$, where $f\left(x\right)={(x—a)}^2$, with $a>1$, is shown below.

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

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

Sketch the graph of $y=—f(—x)$, labelling the images of the points $A$ and $B$ and stating their coordinates in terms of $a$.

Write down the value of $a$ such that the point $A$ is three times as far from the origin as the point $B$.

The diagram shows the graph of $y=f\left(t\right)$, where $f\left(t\right)=\sin 2t,0°\le x\le 180°.$

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

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

(i) Write down the minimum value of $y$ when $y=5f(t+30°).$

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

Find, in terms of $f\left(t\right)$, the combination of transformations that would map the graph of $y=f\left(t\right)$ onto the graph of $y=2+\sin t,0°\le x\le 180°.$

The function $f\left(x\right)$ is to be transformed by a sequence of functions, in the order detailed below:

1. A horizontal stretch by 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 an expression for the combined transformation in terms of $f\left(x\right)$.

The diagram below shows the graph of $y=g\left(x\right)$ where

$g\left(x\right)=\frac{2x+1}{x-1},x\ne 1$

Write down the equations of the two asymptotes.

Determine the equations of the two asymptotes on the graph of $y=g\left(2x\right)-3$.

The point with coordinates $(1,-4)$ is a stationary point on the graph with equation $y=\mathrm{h}\left(x\right)$.

Determine the coordinates of the stationary point on the graphs with the following equations:

(i) $y=2h\left(x-1\right),$

(ii) $y=-h\left(x+1\right)+2.$

The graph of $f$ is shown below. The points $A(-2,10)$ and $B(4,-10)$ lie on the curve.

Sketch the graph of:

(i) $y=f(2x-1),$

(ii) $y=f\left(4-x\right),$

Clearly indicate the new coordinates of the images of the points A and B.

Describe a sequence of transformations that map the graph of $y=\mathrm{In}x$ onto the graph of $y=5+\mathrm{In}\left(\frac{1}{2}x+4\right).$

The diagram below shows the graph of $y=f\left(x\right)$.

The stationary points and intercepts with the $x$-axis are marked on the diagram.

On separate diagrams, sketch the graphs with equations

(i) $y=f\left(\frac{1}{2}x\right)+2,$

(ii) $y=-f\left(x-1\right).$

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

Describe, in order, a sequence of transformations that maps the graph of $y=f\left(x\right)$ onto the following graphs:

(i) $y=-f(3x-1),$

(ii) $y=2f(5-x).$

Given that $f\left(x\right)=\mathrm{In}(2x+1)$ find an expression for $g\left(x\right)$, where $g\left(x\right)$ is obtained by applying the following sequence of transformations to $f\left(x\right)$.

1. Translation by $\left(\begin{array}{c}-3\\ 0\end{array}\right)$,

2. Horizontal stretch by scale factor $\frac{1}{2}$,

3. Reflection in the $x$-axis.

On the same axes sketch the graphs of $y=p\left(x\right)$ and $y=p^{-1}\left(x\right)$, where $p\left(x\right)=\left|2x\right|,x\le 0$.

Find an expression for $p^{-1}\left(x\right)$ and state its domain.

Show that  $p^{-1}\left(x\right)=-\frac{1}{2}p\left(-\frac{1}{2}x\right).$

A sketch of the graph with equation $y=f\left(x\right)$, where $f\left(x\right)={\left(x^2-4\right)}^2$ is shown below.

The points $A,B$ and $C$ are the points where the graph intercepts the coordinate axes.

Sketch the graph of $y=-3f\left(2x\right)$, labelling the images of the three points $A,B$ and $C$.

Suggest a combination of at least two transformations that will transform the points $A,B$ and $C$ such that none of them lie on the coordinate axes. Give your answer in the form of an expression in terms of $f\left(x\right)$.

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

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

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

Find, in terms of $f\left(t\right)$, the combination of transformations that would map the graph of $y=f\left(t\right)$ onto the graph of $y=2-4\sin t,0°\le x\le 180°.$

The function $f\left(x\right)$ is to be transformed by a sequence of functions, in the order detailed below.

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

2. A reflection in the $y$-axis

3. A vertical stretch by scale factor $\frac{2}{3}$

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

Write down the combined transformation in terms of $f\left(x\right)$.

The diagram below shows the graph of $y=g\left(x\right)$ where

$g\left(x\right)=\frac{1}{ax},a,x\ne 0$

where $a$ is a constant.

(i) Write down the equations of the asymptotes on the graph of $y=g\left(x\right)$.

(ii) Determine the equations of the asymptotes on the graph of $y=3g\left(2x+1\right).$

Determine the domain and range of the series of transformations to $y=f\left(x\right)$ where $f\left(x\right)=-2g\left(\frac{1}{3}x+3\right)-4$.

The point with coordinates $(-3,-5)$ is a stationary point on the graph with equation $y=h\left(x\right).$

Determine the coordinates of the stationary point on graphs with the following equations:

(i) $y=h\left(2x\right)-5,$

(ii) $y=h\left(\frac{1}{4}x+1\right),$

(iii) $y=2-\frac{1}{5}h\left(\frac{1}{2}x\right).$

The diagram be10%shows the graph of $y=f\left(x\right)$.

The stationary points and intercepts with the coordinate axes are marked on the diagram.

On separate diagrams, sketch the graphs with the following equations:

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

(ii) $y=-2f\left(x+1\right)+1.$

On each diagram, mark the coordinates of the images of the points $A,B$ and $C$ under the given transformation, giving your coordinates in terms of $a$.

Describe, in order, a sequence of transformations that would map the graph of $y=f\left(x\right)$ onto each of the following graphs:

(i) $y=af\left(x+b\right)+c$     for the case when $a>0$

(ii) $y=-f(—x)$

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

The function $f\left(x\right)=e^{3x}-x-6$ is transformed by a sequence of transformations as described below.

1. Horizontal stretch by scale factor 3,

2. The modulus of the function is then taken,

3. Reflection in the $y$-axis.

Write down the resulting transformation in terms of $f\left(x\right)$ as well as an expression in terms of $x$.

Show that the graph of $y=p\left(x\right)$ where $p\left(x\right)=2x+1$ maps onto the graph of its inverse under the transformations described by  $\frac{1}{2}p\left(\frac{1}{2}x\right)-1$.

Prove, that for a constant $k,k\ne 0,$ if $f\left(x\right)=kx$, then $f^{-1}\left(x\right)=\frac{1}{k}f\left(\frac{1}{k}x\right).$

A sketch of the graph with equation $y=f\left(x\right)$, where $f\left(x\right)={(x^2-a)}^2$, with $a>1$ shown below.

The points $A,B$ and $C$ are points where the graph intercepts the coordinate axes.

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

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

Suggest, in terms of $f\left(x\right)$, a combination of at least two transformations, such that the points $A,B$ and $C$ transform to new positions but remain lying on their respective axes.

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

The vertical distance between the minimum point, $(90,1)$, and the maximum point, $(270,-1)$ is 2. The horizontal distance between them is 180.

Find, in terms of $a$, the vertical and horizontal distances between the minimum and maximum point on the graph of $y=-\frac{1}{a}f\left(at\right),a\ne 0.$

Hence or otherwise show that the distance between the minimum and maximum point on the graph of $y=-\frac{1}{a}f\left(at\right),a\ne 0,$ is

$\frac{2\sqrt{8101}}{a}$

The point with coordinates $(-a,a^2)$, where $a>0$, is a stationary point on the graph with equation $y=h\left(x\right)$. Determine, in terms of a, the coordinates of the stationary point on the graphs with the following equations:

(i) $y=3h\left(\frac{1}{2}x\right)-2,$

(i) $y=3-h\left(-x\right),$

(iii) $y=-\frac{1}{3}\left[h\left(\frac{1}{3}x-\frac{1}{3}\right)+1\right].$