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

The graph of $y=\mathrm{f}\left(x\right)$ is transformed in the following order:

1. A horizontal stretch by scale factor 2

2. A reflection in the $x$-axis

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

Find the equation of the graph after these transformations, in terms of $\mathrm{f}\left(x\right)$.

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

Find the coordinates of the stationary point on the following curves:

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

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

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

The graph of $y=\mathrm{f}\left(x\right)$ is shown below.

The points $A\left(-2,10\right)$ and $B\left(4,-10\right)$ are the stationary points on the curve.

Sketch each of the graphs below.

(i) $y=\mathrm{f}(2x-1)$

(ii) $y=\mathrm{f}(4-x)$

Label the coordinates of the images of the points $A$ and $B$.

Describe, in order, a sequence of transformations that maps the graph of $y=\ln x$ on to the graph of

$y=5+\ln \left(\frac{1}{2}x+4\right)$

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 label the coordinates of the turning point.

(ii) State the vertical 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 shows the graph of $y=\mathrm{f}\left(t\right)$, where

$\mathrm{f}\left(t\right)=\sin \left(2t\right)0°\le t\le 180°$

(i) Write down the maximum value of $y$ on the graph of $y=3\mathrm{f}\left(t\right)$ for $0°\le t\le 180°$.

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

(i) Find the minimum value of $y$ on the graph of $y=5\mathrm{f}(t+30°)$ for $0°\le t\le 180°$.

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

The transformation $\mathrm{f}\left(at\right)+b$ maps $y=\mathrm{f}\left(t\right)$ on to the graph of $y=2+\sin t$, where $0°\le t\le 180°.$

Find the values of $a$ and $b$.

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

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

On separate diagrams, sketch the graphs with equations

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

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

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

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.

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

Sketch the graph of $y=-\mathrm{f}(-x)$

Label the coordinates of the images of the points $A$and $B$ in terms of $a$.

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

The graph $y=\mathrm{f}\left(x\right)$ is transformed in the following 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 by a scale factor of $\frac{2}{3}$

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

Find the equation of the graph after these transformations, in terms of $\mathrm{f}\left(x\right)$.

A function is given by

$\mathrm{f}\left(x\right)=3x^2-2x$

The curve $y=\mathrm{f}\left(x\right)$ is transformed to the curve $y=\mathrm{g}\left(x\right)$ by applying the following transformations to $y=\mathrm{f}\left(x\right)$ in order:

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

2. Vertical stretch of scale factor 4

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

Find an expression for $\mathrm{g}\left(x\right)$ in the form $a{x}^2+bx+c$ where $a$, $b$ and $c$ are constants to be found.

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

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

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

A function is given by

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

A different function, $\mathrm{g}\left(x\right)$, is obtained by applying the following sequence of transformations to $\mathrm{f}\left(x\right)$ in order:

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

2. A horizontal stretch by a scale factor of $\frac{1}{2}$

3. A reflection in the $x$-axis

Find an expression for $\mathrm{g}\left(x\right)$, giving your answer in the form $a\ln \left(bx+c\right)$ where $a$, $b$ and $c$ are integers.

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 meets the coordinate axes.

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

On the graph of $y=\mathrm{f}\left(x+a\right)+b$ where $a$ and $b$ are non-zero constants, none of the points $A$, $B$ and $C$ lie on the coordinate axes.

State the values that $a$ and $b$ cannot take.

The diagram shows the graph of $y=\mathrm{f}\left(t\right)$, where

$\mathrm{f}\left(t\right)=\cos t0°\le t\le 360°$

(i) Find the maximum value of $y$ on the graph of $y=-2\mathrm{f}\left(3t\right)$

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

Find in the form

$a\mathrm{f}\left(bt+c\right)+d$

where $a$, $b$, $c$ and $d$ are constants, a transformation that would map the graph of $y=\mathrm{f}\left(t\right)$ onto the graph of

$y=2-4\sin t0°\le t\le 180°$

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

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

Find the equations of the asymptotes.

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

Find the range of the function $\mathrm{h}$ where

 $\mathrm{h}\left(x\right)=\left|\mathrm{g}\right(3x)-2|x\ne \frac{1}{3}$

The function $\mathrm{g}$ is given by

$\mathrm{g}\left(x\right)=\frac{1}{ax}$

where

• $a$ is a non-zero constant

• the domain of $\mathrm{g}$ is $x\ne 0$

The graph of $y=\mathrm{g}\left(x\right)$ is shown below.

Find the equations of the asymptotes on the graph of $y=3\mathrm{g}(2x+1)$.

The function $\mathrm{f}$ is defined by

$\mathrm{f}\left(x\right)=-2\mathrm{g}\left(3+\frac{x}{3}\right)-4$

• The domain of $\mathrm{f}$ is $x\ne p$

• The range of $\mathrm{f}$ is $\mathrm{f}\left(x\right)\ne q$

Find the constants $p$ and $q$.

The point with coordinates $(-3,-5)$ is the only stationary point on the curve with equation $y=\mathrm{h}\left(x\right)$.

Find the coordinates of the stationary point on the following curves:

(i) $y=\left|\mathrm{h}\right(2x)-5|$

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

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

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 label the coordinates of the local maximum point.

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

A function is given by

$\mathrm{f}\left(x\right)={\mathrm{e}}^{3x}-x-6$

The curve $y=\mathrm{f}\left(x\right)$ is transformed to the curve $y=\mathrm{g}\left(x\right)$ by applying the following transformations to $y=\mathrm{f}\left(x\right)$ in order:

1. A horizontal stretch of scale factor 3

2. The modulus of the function is then taken

3. A reflection in the $y$-axis

Show that

$\mathrm{g}\left(x\right)=a\left|\frac{b}{{\mathrm{e}}^x}+x+c\right|$

where $a$, $b$ and $c$ are constants to be found.

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

The coordinates of the stationary points and intercepts with the axes are also shown, given in terms of $a$ where $a$ is a positive constant.

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

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

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

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

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

$y=a\mathrm{f}(x+b)+c$

where $a$, $b$ and $c$ are constants and $a>1$.

Explain how the answer to part (a) changes in the cases when

(i) $a=1$

(ii) $a<0$

A function is given by

$\mathrm{p}\left(x\right)=2x+1x\in \mathrm{ℝ}$

The graph of $y=\mathrm{p}\left(x\right)$ is transformed in the following order:

1. A horizontal stretch of scale factor $k$, where $k>0$

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

3. A vertical stretch of scale factor $\frac{1}{k}$

The equation of the graph after these transformations is $y={\mathrm{p}}^{-1}\left(x\right)$, where ${\mathrm{p}}^{-1}$ is the inverse function of $\mathrm{p}$.

Find the values of the constants $k$ and $m$.

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 meets the coordinate axes.

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

Sketch the graph of $y=-\frac{1}{2}\mathrm{f}(x-1)$

Label the coordinates of the images of the three points $A$, $B$ and $C$ in terms of $a$.

The graph of

$y=a\mathrm{f}\left(bx+c\right)+d$

is such that

• the images of the points $A$, $B$ and $C$ lie on their original coordinate axes

• the image of $B$ is three times its previous distance from the origin

• the images of $A$ and $B$ are 20% of their previous distances from the origin

• the $x$-coordinate of the point $C$ is negative

Find $a$, $b$, $c$ and $d$.

The point with coordinates $(-a,a^2)$ where $a>0$ is the only stationary point on the curve with equation $y=\mathrm{h}\left(x\right)$.

Find, in terms of $a$, the coordinates of the stationary point on the following curves:

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

(ii) $y=3-\left|\mathrm{h}\right(-x\left)\right|$

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

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.

• The points $A$and $B$ are the $x$-axis intercepts

• The point $C$ is the maximum point

On the same diagram, sketch the graph of

$y=-\left|\frac{1}{4}\mathrm{f}\left(\frac{1}{2}x\right)\right|$

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The area of triangle $ABC$ is $k$ times the area of triangle $A\text{'}B\text{'}C\text{'}$.

Find $k$, giving a reason for your answer.

The diagram shows the graph of $y=\mathrm{f}\left(t\right)$, where

$\mathrm{f}\left(t\right)=\mathrm{cosec}t0°\le t\le 360°$

• The vertical distance between the minimum point and the maximum point is 2

• The horizontal distance between the minimum point and the maximum point is 180

Find, in terms of $a$, the vertical and horizontal distances between the minimum point and maximum point on the graph of

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

The exact distance between the minimum point and maximum point on the graph of

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

is given by

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

where $k$ is a positive integer.

Find the value of $k$.