1.6 Combinations of Transformations

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.

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 .

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. Vertical stretch of scale factor 4
3. Translation by 

Find an expression for .

 Find an expression for .

What can you deduce about the sequence of transformations given
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 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.

Write down the minimum value of  when .

Write down the first value of  for which this minimum 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 .

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  or if ?

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  is 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

The diagram below shows the graph of .

The graph has asymptotes with equations  and

The graph intercepts the -axis at the points A and B.

Write down the coordinates of points  and  .

Determine the coordinates of the points  on  the graph of

Determine the equations of the asymptotes on the graph of

(ii) Find the domain of