Combinations of Transformations (Edexcel IAL Maths): Exam Questions

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

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

On separate diagrams, sketch the following graphs

(i) $y=2f(x)+1$

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

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

Given that $y=f(x)$, find equations, in terms of $f(x)$, for the following transformations:

(i) Translation by $( \begin{gathered} - 2 \\ 3 \end{gathered} )$

(ii) Horizontal stretch of scale factor 2 followed by a vertical stretch of scale factor 3

The graph of $y=f(x)$ where $f(x)=2x-3$ is shown below.

Determine the coordinates of the points marked $A$ and $B$.

(i) On the diagram above sketch the graph of $y = \vert \mathrm{f} ( x - 1 ) \vert$.

(ii) Determine the coordinates of the image of the points $A$and $B$ under the transformation in part (i).

Describe, in order, a sequence of transformations of the graph of $y=f(x)$ given by the following equations:

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

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

The function $g(x)$ is given as $g(x)=2x$.

On the same diagram, sketch the graphs of $y=g(x)$ and $y=g^{-1}(x)$ Label the coordinates of the points where each graph crosses the coordinate axes.

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

(ii) Find an expression for $g^{-1}(x)$ in terms of $g(x)$ and state the type of transformation this would be.

The equation $y=f(x)$, where $f(x)=(x-2)^{2}$, is shown below..

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

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

(ii) The graph of $y=f(x)$ is transformed to the graph of $y=-f(x)-4$. Find the coordinates of the images of points $A$and $B$ under these transformations.

The diagram shows the graph of $y=f(t)$, where $f(t)=\mathrm{cos}t$, $0^{\circ}\leq t\leq360^{\circ}.$

(i) Write down the minimum value of $y=f(t)$ in the given domain for $t$.

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

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

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

Find, in terms of $f(t)$, the combination of transformations that would map the graph of $y=f(t)$ onto the graph of $y=3\mathrm{cos}t+1$, $0^{\circ}\leq t\leq360^{\circ}.$

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

1. A reflection in the $y$-axis.
2. A vertical stretch of scale factor 4.
3. A translation by $( \begin{gathered} 0 \\ - 2 \end{gathered} )$

Write down an expression for the combined transformation in terms of $f(x)$.

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

$g(x)=\frac{1}{x},x\neq0$

(i) Write down the equations of the two asymptotes.

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

The diagram below shows the graph of $y=f(x)$. The stationary points are marked on the diagram.

On separate diagrams, sketch the graphs with equation

(i) $y=2f(x)-4$

(ii) $y=f(x+1)+3$

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 $y=f(x)$ onto the following graphs:

(i) $y=3f(x+2)$,

(ii) $y=f(-x)-1$.

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

1. Translation by $( \begin{gathered} 2 \\ 0 \end{gathered} )$

2. Vertical stretch of scale factor 4

3. Translation by $( \begin{gathered} 0 \\ - 3 \end{gathered} )$

(i) Sketch the graph of $y=p(x)$, where $p(x)=3x-4$.

(ii) On the same set of axes, sketch the graph of $y=p^{-1}(x)$. Label the coordinates of the points where each graph crosses the coordinate axes

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

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

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

The equation $y=f(x)$, where $f(x)=(x-a)^{2}$, 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 $B$ 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 $y=f(t)$, where $f(t)=\mathrm{sin}2t$, $0^{\circ}\leq x\leq180^{\circ}.$

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

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

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

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

Find, in terms of $f(t)$, the combination of transformations that would map the graph of $y=f(t)$ onto the graph of $y=2+\mathrm{sin}t$, $0^{\circ}\leq x\leq180^{\circ}$ .

The function $f(x)$ 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 $( \begin{gathered} 0 \\ 2 \end{gathered} )$

Write down an expression for the combined transformation in terms of $f(x)$.

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

$g(x)=\frac{2x+1}{x-1},x\neq1$

Write down the equations of the two asymptotes.

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

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

The point with coordinates $(1,-4)$ is a stationary point on the graph with equation $y=h(x)$.

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

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

(ii) $y=-h(x+1)+2$,

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

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

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

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

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

On separate diagrams, sketch the graphs with equations

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

(ii) $y=-f(x-1)$

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

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

1. Translation by $( \begin{gathered} - 3 \\ 0 \end{gathered} )$,
2. Horizontal stretch by scale factor $\frac{1}{2}$,
3. Reflection in the $x$-axis.

On the same axes sketch the graphs of $y=p(x)$ and $y=p^{-1}(x)$, where $\mathrm{p} ( x ) = \vert 2 x \vert , x \text{less} \text{or} \text{equal} \text{than} 0$.

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

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

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

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

Sketch the graph of $y=-3f(2x)$, 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 $f(x)$.

The diagram shows the graph of $y=f(t)$, where $f(t)=\mathrm{cos}t$, $0^{\circ}\leq x\leq360^{\circ}.$

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

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

Find, in terms of $f(t)$, the combination of transformations that would map the graph of $y=f(t)$ onto the graph of $y=2-4\mathrm{sin}t$,$0^{\circ}\leq x\leq180^{\circ}$.

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

1. A translation by $( \begin{gathered} 2 \\ 0 \end{gathered} )$
2. A reflection in the $y$-axis
3. A vertical stretch by scale factor $\frac{2}{3}$
4. A translation by $( \begin{gathered} 0 \\ 4 \end{gathered} )$

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

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

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

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

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

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

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

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

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

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

Prove, that for a constant $k,k\neq0$, if $f(x)=kx$, then $f^{-1}(x)=\frac{1}{k}f(\frac{1}{k}x)$.

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

Suggest, in terms of $f(x)$, 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 $y=f(t)$, where $f(t)=\mathrm{cosec}t$, $0^{\circ}\leq x\leq360^{\circ}.$

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 $y=-\frac{1}{a}f(at),a\neq0$.

Hence or otherwise show that the distance between the minimum and maximum point on the graph of $y=-\frac{1}{a}f(at),a\neq0$, i

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

A sketch of the graph with equation $y=f(x)$ where $f(x)=10x-x^{2}-16$ 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 $y = - \vert 1 \frac{1}{4} \mathrm{f} ( 1 \frac{1}{2} x ) \vert$labelling the image of the points $A,B$ and $C$ with $A^{'},B^{'}$and $C'$.

Show that the area of $ABC$ is twice the area of triangle $A'B'C'$.