2.10 Combinations of Transformations (A Level only)

Describe, in order, a sequence of transformations that maps the graph of  onto the following graphs:

Given that  find an expression for , where  is obtained by applying the following sequence of transformations to .

1. Translation by ,
2. Horizontal stretch by scale factor ,
3. Reflection in the -axis.

On the same axes sketch the graphs of  and , where .

Find an expression for  and state its domain.

Show that 

A sketch of the graph with equation , where  is shown below.

The points  and  are the points where the graph intercepts the coordinate axes.

Sketch the graph of , labelling the images of the three points  and .

Suggest a combination of at least two transformations that will transform the points and  such that none of them lie on the coordinate axes.
Give your answer in the form of an expression in terms of .

The diagram shows the graph of , where

Write down the maximum value of when 

Write down the value of for which this maximum occurs.

Find, in terms of , the combination of transformations that would map the graph of  onto the graph of 

The function  is to be transformed by a sequence of functions, in the order detailed below.

1. A translation by 
2. A reflection in the -axis
3. A vertical stretch by scale factor 
4. A translation by 

Write down the combined transformation in terms of .

The diagram below shows the graph of  where

where  is a constant.

Write down the equations of the asymptotes on the graph of .

Determine the equations of the asymptotes on the graph of

Determine the domain and range of the series of transformations to where .

The point with coordinates  is a stationary point on the graph with equation 

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

Describe, in order, a sequence of transformations that maps the graph of onto the following graphs:

Given that  find an expression for , where  is obtained by applying the following sequence of transformations to .

Translation by

Vertical stretch of scale factor

Translation by

The equation , where , with , is shown below.

The points  and  are the points where the graph intercepts the coordinate axes.

Write down, in terms of , the coordinates of and .

Sketch the graph of , labelling the images of the points  and  and stating their coordinates in terms of .

Write down the value of  such that the point  is three times as far from the origin as the point .

The diagram shows the graph of , where 

Write down the maximum value of when

Write down the first value of for which this maximum occurs.

Find, in terms of , the combination of transformations that would map the graph of  onto the graph of

The function  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 -axis
3. A translation by 

Write down an expression for the combined transformation in terms of .

The diagram below shows the graph of  where

Write down the equations of the two asymptotes.

Determine the equations of the two asymptotes on the graph of .

The point with coordinates  is a stationary point on the graph with equation .

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

The graph of  is shown below. The points  and  lie on the curve.

Sketch the graph of:

Clearly indicate the new coordinates of the images of the points A and B.

Describe a sequence of transformations that map the graph of  onto the graph of 

Describe, in order, a sequence of transformations that would map the graph of onto each of the following graphs:

     for the case when

How, if at all, would your answer to part (a) (i) change if 

The function  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 -axis.

Write down the resulting transformation in terms of  as well as an expression in terms of .

Show that the graph of  where  maps onto the graph of its inverse under the transformations described by  .

Prove, that for a constant if , then 

A sketch of the graph with equation , where , with  shown below.

The points  and  are points where the graph intercepts the coordinate axes.

Write down, in terms of , the coordinates of  and .

Sketch the graph of , labelling the images of the three points and  and stating their coordinates in terms of .

Suggest, in terms of , a combination of at least two transformations, such that the points  and  transform to new positions but remain lying on their respective axes.

The diagram shows the graph of , where

The vertical distance between the minimum point, , and the maximum point, is 2. The horizontal distance between them is 180.

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

Hence or otherwise show that the distance between the minimum and maximum point on the graph of is

The point with coordinates , where , is a stationary point on the graph with equation .
Determine, in terms of a, the coordinates of the stationary point on the graphs with the following equations: