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

The point $P(0,5)$ lies on the curve with equation $y=\mathrm{f}\left(x\right)$.

Find the coordinates of the image of the point $P$ on the curves with the following equations:

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

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

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The points $A(0,0)$ and $B(4,8)$ are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

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

(ii) $6y=\mathrm{f}\left(x\right)$

On each diagram, label the coordinates of the stationary points.

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

The point $P$ has coordinates $\left(a,b\right)$, where $a,b>0$

Giving your answers in terms of $a$ and $b$, find the coordinates of the image of the point $P$ under the following graph transformations:

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

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

(iii) $y=a\mathrm{f}\left(x\right)$

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The curve intersects the coordinate axes at the two points, $A(0,6)$ and $B(3,0)$. 

The curve has two asymptotes, as shown, with equations $y=\frac{20}{3}$  and $x=\frac{10}{3}$.

On separate diagrams, sketch the curves with equation

(i) $y=\mathrm{f}\left(5x\right)$

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

On each diagram, label the coordinates of the images of points $A$ and $B$ under the given transformation and the equations of any asymptotes.

The graph of $y=a\mathrm{f}\left(x\right)$, where $a$ is a constant, has an asymptote with equation $y=2$. 

Find the value of $a$.

The function $\mathrm{f}\left(x\right)$ is defined by

 $\mathrm{f}\left(x\right)=(x-2)(x-6)$

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

On your sketch, show clearly

• the coordinates of the points where the graph intersects the coordinate axes

• the coordinates of the turning point

On separate diagrams, sketch the graphs of:

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

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

In each case, show clearly

• the coordinates of the points where the graph intersects the coordinate axes

• the coordinates of the turning point

The point $P(3,2)$ lies on the curve with equation $y=\mathrm{f}\left(x\right).$

(i) On the graph of $y=\mathrm{f}\left(x\right)+a$, where $a$is a constant, the point $P$ is mapped to the point $(3,-5)$. Determine the value of  $a$.

(ii) On the graph of $y=\mathrm{f}(x+b)$, where $b$is a constant, the point $P$ is mapped to the point $(-1,2)$. Determine the value of $b$.

(iii) On the graph of $y=c\mathrm{f}\left(x\right)$, where $c$is a constant, the point $P$is mapped to the point $(3,1)$. Determine the value of $c$.

(iv) On the graph of $y=\mathrm{f}\left(dx\right)$, where $d$is a constant, the point $P$ is mapped to the point $(1,2)$. Determine the value of $d$.

The curve with equation $y=\mathrm{f}\left(x\right)$ has two asymptotes with equations $y=-3$ and $x=2$.

Find the equations of the asymptotes for the following curves:

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

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

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The points $A(-1,5)$ and $B(3,-3)$ are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

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

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

On each diagram, label the coordinates of the stationary points.

On the graph of $y=\mathrm{f}(x+a)$, where $a$ is a constant, the $x$ coordinate of one of the stationary points is 2. 

Find the possible values of $a$.

The curve with equation $y=\mathrm{f}\left(x\right)$ has two asymptotes with equations $y=5$and $x=-4.$ 

Find the equations of the asymptotes for the following curves:

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

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

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The points $A(0,0)$ and $B(4,8)$ are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

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

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

On each diagram, label the coordinates of the stationary points.

On the graph of $y=a\mathrm{f}\left(x\right)$, where $a$ is a constant, the $y$ coordinate of one of the stationary points is 4.

Find the value of $a$.

The curve with equation $y=\mathrm{f}\left(x\right)$ has two asymptotes with equations $y=-1$ and $x=-2$. 

Find the equations of the asymptotes for the following curves:

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

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

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The curve intersects the coordinate axes at the two points, $A(0,6)$ and $B(3,0)$. 

The curve has two asymptotes, as shown, with equations $y=\frac{20}{3}$  and $x=\frac{10}{3}$.

On separate diagrams, sketch the curves with equation

(i) $y=\mathrm{f}\left(x\right)-6$

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

On each diagram, label the coordinates of the images of points $A$ and $B$ under the given transformation and the equations of any asymptotes.

The curve with equation $y=\mathrm{f}(x+a)$ has an asymptote along one of the coordinate axes. 

Find the value of $a$.

The point $P(3,-12)$ lies on the curve $C$ with equation $y=x^2-12x+15$.

The curve $C$ is stretched so that the point $P$ is mapped to the point $(3,-4)$.

Find the equation of the transformed curve.

Give your answer in the form $y=a{x}^2+bx+c$, where $a$,$b$  and $c$ are constants to be found.

If, instead, the curve $C$ is stretched so that the point $P$is mapped to the point $(1,-12)$, find the equation of the transformed curve.

Give your answer in the form $y=(dx{)}^2-12(dx)+15$, where $d$ is a constant to be found.

Sketch the curve

$y=\frac{1}{x}+3$

Label clearly the coordinates of any points where the curve crosses the coordinate axes and give the equations of any asymptotes.

The curve with equation

$y=\frac{1}{\left(x+a\right)}+3$ 

passes through the origin. 

Find the value of $a$.

Given that 

$x^3-10x^2-24x\equiv x(x+2)(x-12)$

sketch the curve with equation $y=x^3-10x^2-24x,$ showing clearly the coordinates of the points where the curve crosses the coordinate axes.

The curve with equation 

$y=(x+a{)}^3-10(x+a{)}^2-24(x+a)$ 

passes through the point $(-2,0)$. 

Find the three possible values of $a$.

The point $P(-3,-2)$ lies on the curve with equation $y=\mathrm{f}\left(x\right)$. 

Find the coordinates of the image of the point $P$ on the curves with the following equations:

(i) $y-2=\mathrm{f}\left(x\right)-6$

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

(iii) $2y=\mathrm{f}\left(x\right)$

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

The point $P(-12,-9)$ lies on the curve $C$ with equation $y=x^2+15x+27$.

The curve $C$ is translated so that the point $P$ is mapped to the point $(-12,3)$. 

Find the equation of the transformed curve.

If, instead, the curve $C$ is translated so that the point $P$is mapped to the point $(-10,-9)$, find the equation of the transformed curve.

Give your answer in the form $y=(x+a{)}^2+15(x+a)+27$, where $a$ is a constant to be found.

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The points $A(-1,5)$ and $B(3,-3)$ are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

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

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

On each diagram, label the coordinates of the stationary points.

On the graph of $y=\mathrm{f}(x+a)$, the two stationary points both lie on the same side of the $y$-axis. 

Find all the possible values that $a$ can take.

Sketch the curve with equation

 $y=2-\frac{8}{x^2}$

Label clearly the coordinates of any points where the curve crosses the coordinate axes and give the equations of any asymptotes.

The curve with equation

$y=2-\frac{8}{{\left(x+a\right)}^2}$ 

passes through the origin. 

Find the two possible values of $a$.

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The points $A(-1,5)$ and $B(3,-3)$ are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

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

(ii) $5y=\mathrm{f}\left(x\right)$

On each diagram, label the coordinates of the stationary points.

On the graph of $y=\mathrm{f}\left(ax\right)$, where $a$  is a constant, the $x$ coordinate of one of the stationary points is $\frac{5}{3}$.

Given that $a>0$, find the value of $a$.

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The points $A(0,0)$ and $B(4,8)$ are the stationary points on the curve.

Consider the three following graph transformations

$y=\mathrm{f}(-x)y=\mathrm{f}\left(ax\right)y=\mathrm{f}\left(x\right)+b$

where $a$ and $b$ are constants and $a>0$.

State which of the transformations satisfies each of the following conditions, and determine the range of possible values of $a$ and $b$ where necessary.

(i) The images of the stationary points under the transformation lie on opposite sides of the $x$-axis.

(ii) The image of point $B$ under the transformation has coordinates $(x,y)$, where $-6<x<-3$.

(iii) The image of point $B$ under the transformation has coordinates $(x,y)$, where $0<x<3$.

.

The diagram below shows the curve with equation $y=\mathrm{f}\left(x\right)$.

The curve intersects the coordinate axes at the two points, $A(0,6)$ and $B(3,0)$. 

The curve has two asymptotes, as shown, with equations $y=\frac{20}{3}$  and $x=\frac{10}{3}$.

On separate diagrams, sketch the curves with equation

(i) $y=\mathrm{f}\left(\frac{20}{3}x\right)$

(ii) $5y=4\mathrm{f}\left(x\right)$

On each diagram, label the coordinates of the images of points $A$ and $B$ under the given transformation and the equations of any asymptotes.

The graph of $y=\mathrm{f}\left(ax\right)$, where $a$ is a constant, has an asymptote with equation $x=k$, where $1<k<100$. 

Find the range of possible values of $a$.

Given that 

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

sketch the graph of $y=\mathrm{f}\left(x\right)$, showing clearly the coordinates of the points where the curve crosses or touches the coordinate axes.

The functions $\mathrm{g}\left(x\right)$ and $\mathrm{h}\left(x\right)$ are defined by

$\begin{array}{l}\mathrm{g}\left(x\right)=\mathrm{f}(-x)\\ \mathrm{h}\left(x\right)=\mathrm{g}(x+a)\end{array}$

The graph of $\mathrm{h}\left(x\right)$ touches the $x$-axis at the point $(5,0)$. 

Find the exact value of $a$.

The function $\mathrm{f}\left(x\right)$ is defined by

$\mathrm{f}\left(x\right)=9-\frac{16}{{\left(x-2\right)}^2}$

Sketch the graph of $y=\mathrm{f}\left(x\right)$, showing clearly

• the coordinates of the points where the curve crosses the coordinate axes

• the equations of any asymptotes

The graph of $y=\mathrm{f}(x+a)$  is such that any point on the graph with a $y$-coordinate of less than 5 has a negative $x$-coordinate.

Find the range of possible values of $a$.