Transformations of Functions (Cambridge International AS Mathematics): Exam Questions

A curve has equation $y = \text{f} ( x )$.

Describe fully the single transformation that maps the graph of $y = \text{f} ( x )$ onto the graph of each of the following:

(i) $y = \text{f} ( x ) + 2$

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

(iii) $y = 3 \text{f} ( x )$

(iv) $y = \text{f} ( 2 x )$

A curve has equation $y = \text{f} ( x )$.

Write down, in terms of $\text{f}$, the equation of the curve obtained by each of the following transformations of $y = \text{f} ( x )$:

(i) a translation by \begin{pmatrix} 3 \\ 0 \end{pmatrix}

(ii) a stretch parallel to the $x$-axis with scale factor $2$

(iii) a stretch parallel to the $y$-axis with scale factor $\frac{1}{3}$

(iv) a reflection in the $y$-axis

The point P \left( 2, 6 \right) lies on the curve with equation $y = \text{f} ( x )$.

State the coordinates of the image of $P$ on each of the following curves:

(i) $y = \text{f} ( x ) + 1$

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

(iii) y = \text{f}\left( \tfrac{1}{4}x \right)

A point P \left( -2, 8 \right) on the graph of $y = \text{f} ( x )$ is mapped to the point $P '$ under a single transformation.

For each of the following coordinates of $P '$, describe a transformation that could have produced it:

(i) P' \left( -2, 3 \right)

(ii) P' \left( -4, 8 \right)

(iii) P' \left( -2, -8 \right)

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

Find the value of $a$ in each of the following cases:

(i) on the graph of $y = \text{f} ( x + a )$, the point $P$ is mapped to \left( -3, -4 \right)

(ii) on the graph of $y = a \text{f} ( x )$, the point $P$ is mapped to \left( 3, -12 \right)

(iii) on the graph of $y = \text{f} ( a x )$, the point $P$ is mapped to \left( -3, -4 \right)

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

$\text{f} ( x ) = ( x - 2 ) ( x - 6 )$

Sketch the graph of $y = \text{f} ( x )$, showing clearly the coordinates of the points where the graph meets the coordinate axes and the coordinates of the turning point.

On separate diagrams sketch the graphs of:

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

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

In each case show clearly the coordinates of the points where the graph meets the coordinate axes and the coordinates of the turning point.

The diagram shows the graph of $y = \text{f} ( x )$. The point $P$ has coordinates \left( a, b \right), where $a , b > 0$.

In terms of $a$ and $b$, write down the coordinates of the image of $P$ under each of the following transformations:

(i) $y = \text{f} ( 2 x )$

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

(iii) $y = a \text{f} ( x )$

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

State the coordinates of the image of $P$ on each of the following curves:

(i) $y = \text{f} ( x ) + 3$

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

(iii) $y = 3 \text{f} ( x )$

(iv) $y = \text{f} ( 3 x )$

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

State the coordinates of the image of $P$ on each of the following curves:

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

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

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

Determine the value of the constant in each of the following cases:

(i) on the graph of $y = \text{f} ( x ) + a$, the point $P$ is mapped to \left( 3, -5 \right)

(ii) on the graph of $y = \text{f} ( x + b )$, the point $P$ is mapped to \left( -1, 2 \right)

(iii) on the graph of $y = c \text{f} ( x )$, the point $P$ is mapped to \left( 3, 1 \right)

(iv) on the graph of $y = \text{f} ( d x )$, the point $P$ is mapped to \left( 1, 2 \right)

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

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ on each:

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

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

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

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

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ on each:

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

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

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

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

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ and the equations of the transformed asymptotes:

(i) $y = \text{f} ( x ) - 6$

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

On the graph of $y = \text{f} ( x + a )$ one of the asymptotes lies on a coordinate axis. 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 $y = \frac{1}{x+a} + 3$ passes through the origin. Find the value of $a$.

Given that

$x^{3} - 10 x^{2} - 24 x = x ( x + 2 ) ( x - 12 )$

sketch the graph of $y = x^{3} - 10 x^{2} - 24 x$, 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 \left( -3, -2 \right) lies on the curve with equation $y = \text{f} ( x )$.

State the coordinates of the image of $P$ on each of the following curves:

(i) $y - 2 = \text{f} ( x ) - 6$

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

(iii) $2 y = \text{f} ( x )$

(iv) y = \text{f}\left( \tfrac{1}{2}x \right)

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

State the coordinates of the image of $P$ on each of the following curves:

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

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

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

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ on each:

(i) y = \text{f}\left( \tfrac{1}{3}x \right)

(ii) $6 y = \text{f} ( x )$

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

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ and the equations of the transformed asymptotes:

(i) $y = \text{f} ( 5 x )$

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

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

Given that

$x^{3} - 8 x^{2} + 16 x = x ( x - 4 )^{2}$

sketch the graph of $y = x^{3} - 8 x^{2} + 16 x + 3$, showing clearly the coordinates of the point where the curve crosses the $y$-axis and the coordinates of any minimum points.

The curve crosses the $x$-axis once; show this crossing in the correct position, but you are not required to find its coordinates.

(You do not need to state the coordinates of any maximum points.)

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

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

Give the equations of the asymptotes of each of the following curves:

(i) $y + 3 = \text{f} ( x )$

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

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

Give the equations of the asymptotes of each of the following curves:

(i) $\frac{1}{3} y = \text{f} ( x )$

(ii) y = \text{f}\left( \tfrac{1}{3}x \right)

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

Give the equations of the asymptotes of each of the following curves:

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

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

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

Consider the three transformations of the graph

$y = \text{f} ( - x )$

$y = \text{f} ( a x )$

$y = \text{f} ( x ) + b$

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

State which of the transformations satisfies each of the following conditions, determining the range of possible values of $a$ or $b$ where relevant:

(i) the images of the two marked points lie on opposite sides of the $x$-axis;

(ii) the image of point $B$ has coordinates \left( x, y \right), where $- 6 < x < - 3$;

(iii) the image of point $B$ has coordinates \left( x, y \right), where $0 < x < 3$.

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

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

The graph is translated so that $P$ is mapped to the point \left( -10, -9 \right). Write down the equation of the transformed curve in the form

$y = ( x + a )^{2} + 15 ( x + a ) + 27$

where $a$ is a constant to be found.

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

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

The graph is stretched so that $P$ is mapped to the point \left( 1, -12 \right). Write down the equation of the transformed curve in the form

$y = ( d x )^{2} - 12 ( d x ) + 15$

where $d$ is a constant to be found.

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

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ on each:

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

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

On the graph of $y = \text{f} ( x + a )$ 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 shows the graph of $y = \text{f} ( x )$. The marked points are A \left( -1, 5 \right) and B \left( 3, -3 \right).

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ on each:

(i) y = \text{f}\left( \tfrac{1}{3}x \right)

(ii) $5 y = \text{f} ( x )$

On the graph of $y = \text{f} ( a x )$ 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 shows the graph of $y = \text{f} ( x )$. The graph meets the coordinate axes at the marked points A \left( 0, 6 \right) and B \left( 3, 0 \right), and has asymptotes with equations $y = \frac{20}{3}$ and $x = \frac{10}{3}$.

On separate diagrams, sketch the following curves, giving the coordinates of the images of $A$ and $B$ and the equations of the transformed asymptotes:

(i) y = \text{f}\left( \tfrac{20}{3}x \right)

(ii) $5 y = 4 \text{f} ( x )$

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

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

$\text{f} ( x ) = 9 - \frac{16}{(x-2)^{2}}$

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

The graph of $y = \text{f} ( x + a )$ is such that, for all points P \left( x, y \right) 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 $a$.

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

$\text{f} ( x ) = x^{3} - 2 \sqrt{3} x^{2} + 3 x$

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

The functions $\text{g}$ and $\text{h}$ are defined by

$\text{g} ( x ) = \text{f} ( - x )$

$\text{h} ( x ) = \text{g} ( x + a )$

The graph of $y = \text{h} ( x )$ touches the $x$-axis at the point \left( 5, 0 \right). Find the exact value of $a$.