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

The point $P(-1,4)$ lies on the curve with equation $y=\mathrm{f}\left(x\right).$. 

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

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

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

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

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

The point $P(-3,-4)$ lies on the curve with equation $y=\mathrm{f}\left(x\right).$.  

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

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

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

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 diagram below shows the graph of $y=\mathrm{f}\left(x\right).$The two marked points $A(-1,5)$ and $B(3,-3)$ lie on the graph.

In separate diagrams, sketch the curves with equation

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

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

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

On the graph of $y=\mathrm{f}(x+a)$ the image of one of the two marked points has an  $x$ coordinate of 2.  Find the two possible values of $a$.

The diagram below shows the graph of $y=\mathrm{f}\left(x\right).$.  The marked point $B(4,8)$ lies on the graph, and the graph meets the origin at the marked point $A$.

In separate diagrams, sketch the curves with equation

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

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

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

On the graph of $y=a\mathrm{f}\left(x\right)$ the image of one of the two marked points has a $y$coordinate of 4. Find the value of $a$.

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$.  The graph intersects the coordinate axes at the two marked points $A(0,6)$ and $B(3,0)$.  The graph has two asymptotes as shown, with equations  $y=\frac{20}{3}$   and  $x=\frac{10}{3}$

In separate diagrams, sketch the curves with equation

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

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

On each diagram, give the coordinates of the images of points $A$ and $B$ under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of $y=\mathrm{f}(x+a)$ has an asymptote at one of the coordinate axes.  Find the value of $a$.

Sketch the graph of  $y=\frac{1}{x}+3$, showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

The graph of $y=\frac{1}{\left(x+a\right)}+3$  passes through the origin.  Find the value of $a$.

Given that $x^3-10x^2-24x=x(x+2)(x-12)$

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

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

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

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

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

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

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

The graph is translated so that the point $P$is mapped to the point $(-12,3)$.  Write down the equation of the transformed function.

The graph is translated so that the point $P$is mapped to the point $(-10,-9)$.  Write down the equation of the transformed function in the form $y=(x+a{)}^2+15(x+a)+27$, where $a$ is a constant to be found.

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

The graph is stretched so that the point $P$is mapped to the point $(3,-4).$.  Write down the equation of the transformed function in the form $y=a{x}^2+bx+c$, where $a$,$b$  and $c$ are constants to be found.

The graph is stretched so that the point $P$is mapped to the point $(1,-12)$. Write down the equation of the transformed function in the form $y=(dx{)}^2-12(dx)+15$, where $d$ is a constant to be found.

The diagram below shows the graph of $y=f\left(x\right)$.  The two marked points $A(-1,5)$ and $B(3,-3)$ lie on the graph.

In separate diagrams, sketch the curves with equation

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

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

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

On the graph of $y=\mathrm{f}(x+a)$ the images of the two marked points both lie on the same side of the $y$-axis.  Find the range of possible values of $a$.

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$. The marked point $B(4,8)$ lies on the graph, and the graph meets the origin at the marked point $A$.

In 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, give the coordinates of the images of points $A$ and $B$ under the given 

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$. The graph intersects the coordinate axes at the two marked points $A(0,6)$ and $B(3,0)$.  The graph has two asymptotes as shown, with equations  $y=\frac{20}{3}$   and  $x=\frac{10}{3}$ .

In separate diagrams, sketch the curves with equation

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

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

On each diagram, give the coordinates of the images of points $A$ and $B$ under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of $y=a\mathrm{f}\left(x\right)$ has an asymptote with equation $y=2$.  Find the value of $a$.

Sketch the graph of  $y=2-\frac{8}{x^2}$, showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

The graph of  $y=2-\frac{8}{{\left(x+a\right)}^2}$  passes through the origin.  Find the two possible values of $a$.

Given that $x^3-8x^2+16x=x(x-4{)}^2$

Sketch the graph of $y=x^3-8x^2+16x+3$ , showing clearly the coordinates of the points where the curve crosses the coordinate axes and the co-ordinates of any minimum points. (You do not need to state the co-ordinates of any maximum points).

The graph with equation   $y+a=x^3-8x^2+16x$  crosses the $x$-axis three times. 
Find the range of possible values of $a$.

The curve with equation $y=\mathrm{f}\left(x\right)$ has two asymptotes, for which the equations are

$y=-3$ and $x=2$. 

Give the equations of the asymptotes for the curves with the following equations:

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

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

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

Give the equations of the asymptotes for the curves with the following equations:

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

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

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

Give the equations of the asymptotes for the curves with the following equations:

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

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

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$.  The two marked points $A(-1,5)$ and $B(3,-3)$ lie on the graph.

In 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, give the coordinates of the images of points $A$ and $B$ under the given transformation.

On the graph of $y=\mathrm{f}\left(ax\right)$ the image of one of the two marked points has an $x$ coordinate of $\frac{5}{3}$.  Given that $a>0$, find the value of $a$.

The diagram below shows the graph of $y=\mathrm{f}\left(x\right)$.  The marked point $B(4,8)$ lies on the graph, and the graph meets the origin at the marked point $A$.

Consider the three following transformations of the graph

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

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 the variables $a$ and $b$ where relevant.

(i) The images of the two marked 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 graph of $y=\mathrm{f}\left(x\right)$.  The graph intersects the coordinate axes at the two marked points $A(0,6)$ and $B(3,0)$.  The graph has two asymptotes as shown, with equations   $y=\frac{20}{3}$   and   $x=\frac{10}{3}$  .

In 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 give the coordinates of the images of points $A$ and B under the given transformation, as well as stating the equations of the transformed asymptotes.

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

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

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

Sketch the graph of $y=\mathrm{f}\left(x\right)$, showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

The graph of $y=\mathrm{f}(x+a)$  is such that, for all points $P(x,y)$ that lie on the graph, if the $y$ coordinate is less than 5 then the $x$ coordinate is less than zero.  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 the equations

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

The graph of $\mathrm{h}\left(x\right)$ touches the $x$-axis at the point $(5,0)$.  Find the value of $a$, giving your answer as an exact value.