Combinations of Transformations (Edexcel International A Level (IAL) Maths: Pure 3): 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) $y=\mathrm{f}(x+1)+3$

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 $g\left(x\right)$, where $g\left(x\right)$ is obtained by applying the following sequence of transformations to $\mathrm{f}\left(x\right)$.

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

   2.   Vertical stretch of scale factor 4

   3.   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 ${\mathrm{p}}^{-1}\left(x\right)$.

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

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

The equation $y=\mathrm{f}\left(x\right)$, where $\mathrm{f}\left(x\right)=(x-a{)}^2$, 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 $y=-\mathrm{f}(-x)$, labelling the images of the points $A$and $B$ stating their coordinates in terms of $a$.

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

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

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

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

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

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

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

The function $\mathrm{f}\left(x\right)$ 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 $\mathrm{f}\left(x\right)$.

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

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

Write down the equations of the two asymptotes.

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

Determine the range of $\left|\mathrm{g}\right(3x)-2|$.

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) $y=2\mathrm{h}(x-1)$,

(ii) $y=-\mathrm{h}(x+1)+2$,

(iii) $y=\left|\mathrm{h}\right(3x)+2|$.

The turning point on the graph of $y=\mathrm{f}\left(x\right)$ has coordinates $(2,-5)$ as shown on the diagram below.

(i) On the diagram above sketch the graph of $y=\left|\mathrm{f}\right(x\left)\right|+1$  and state the coordinates of the turning point.

(ii) State the distance between the turning points on the graphs of $y=\mathrm{f}\left(x\right)$ and $y=\left|\mathrm{f}\right(x\left)\right|+1$. .

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$ . The stationary points and intercepts with the $x$-axis are marked on the diagram.

On separate diagrams, sketch the graphs with equations

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

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

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

Given that $\mathrm{f}\left(x\right)=\ln (2x+1)$ 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 $\mathrm{f}\left(x\right)$.

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 $y=\mathrm{p}\left(x\right)$ and $y=p^{-1}\left(x\right)$, where $\mathrm{p}\left(x\right)=\left|2x\right|,x\le 0$.

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

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

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

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

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

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

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

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

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

The function $\mathrm{f}\left(x\right)$ 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 $\mathrm{f}\left(x\right)$.

The minimum point on the graph of $y=\mathrm{f}\left(x\right)$ has coordinates $(4,-8)$ as shown on the diagram below.

Sketch the graph of $y=\left|\mathrm{f}\right(2x\left)\right|-3$ and state the coordinates of the maximum point.

Find the exact distance between the minimum point on the graph of $y=\mathrm{f}\left(x\right)$ and the maximum point on the graph of $y=\left|\mathrm{f}\right(2x\left)\right|-3$.

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

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

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

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

The function $\mathrm{f}\left(x\right)=e^{3x}-x-6$ 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 $\mathrm{f}\left(x\right)$ as well as an expression in terms of $x$.

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

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

A sketch of the graph with equation $y=\mathrm{f}\left(x\right)$, where $\mathrm{f}\left(x\right)=(x^2-a{)}^2$, with $a>1$ is 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 $y=-\frac{1}{2}\mathrm{f}(x-1)$, labelling the images of the three points $A,B$and $C$and stating their coordinates in terms of $a$.

Suggest, in terms of $\mathrm{f}\left(x\right)$, 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 $y=\mathrm{f}\left(t\right)$, where $\mathrm{f}\left(t\right)=\mathrm{cosec}t$, $0°\le x\le 360°.$

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 $y=-\frac{1}{a}\mathrm{f}\left(at\right),a\ne 0$.

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

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

A sketch of the graph with equation $y=\mathrm{f}\left(x\right)$ where $\mathrm{f}\left(x\right)=10x-x^2-16$ is shown below.

Points $A$and $B$ are the $x$-axis intercepts and point $C$ is the maximum point on the graph.

On the diagram above, sketch the graph of $y=-\left|\frac{1}{4}\mathrm{f}\right(\frac{1}{2}x\left)\right|$labelling the image of the points $A,B$ and $C$ with $A^{\text{'}},B^{\text{'}}$and $C\text{'}$.

Show that the area of $ABC$ is twice the area of triangle $A\text{'}B\text{'}C\text{'}$.