Combinations of Transformations (Edexcel International A Level Maths: Pure 3)

The diagram below shows the graph of .
The stationary points are marked on the diagram.

On separate diagrams, sketch the graphs with equation

On each diagram, state the coordinates of the images of the points and  under the given transformation.

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 .

(ii)     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 .

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 .

Determine the range 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 turning point on the graph of  has coordinates  as shown on the diagram below.

On the diagram above sketch the graph of  and state the coordinates of
the turning point.

State the distance between the turning points on the graphs of and .
.

The diagram below shows the graph of  .
The stationary points and intercepts with the -axis are marked on the diagram.

On separate diagrams, sketch the graphs with equations

(i)     

On each diagram, mark the coordinates of the images of the points and  under the given transformation.

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 minimum point on the graph of  has coordinates  as shown on the diagram below.

Sketch the graph of  and state the coordinates of the maximum point.

Find the exact distance between the minimum point on the graph of  and the maximum point on 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  or if ?

The function  is transformed by a sequence of transformations as described below.

1. 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  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 , i                     

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

Points and  are the -axis intercepts and point  is the maximum point on the graph.

On the diagram above, sketch the graph of labelling the image of the points  and  with and .

Show that the area of is twice the area of triangle .