Combinations of Transformations (Edexcel A Level Maths: Pure): Exam Questions

The point lies on the curve with equation

Find the point to which is mapped, when the curve with equation is transformed to the curve with equation

The diagram shows the graph of , where

(i) Write down the maximum value of  on the graph of for .

(ii) Write down the value of  for which this maximum value occurs.

(i) Find the minimum value of  on the graph of for .

(ii) Find the value of  for which this minimum value occurs.

The transformation maps on to the graph of , where

Find the values of and .

The equation , where , with , is shown below.

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

Find, in terms of , the coordinates of and .

Sketch the graph of

Label the coordinates of the images of the points and in terms of .

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

A function is given by

The curve  is transformed to the curve by applying the following transformations to in order:

1. Translation by 

2. Vertical stretch of scale factor 4

3. Translation by 

Find an expression for in the form where , and are constants to be found.

The graph of  is transformed in the following order:

1. A horizontal stretch by scale factor 2

2. A reflection in the -axis

3. A translation by 

Find the equation of the graph after these transformations, in terms of .

The point with coordinates  is the only stationary point on the curve with equation .

Find the coordinates of the stationary point on the following curves:

(i)

(ii)

(iii)

The graph of is shown below.

The points and are the stationary points on the curve.

Sketch each of the graphs below.

(i)

(ii)

Label the coordinates of the images of the points and .

Describe, in order, a sequence of transformations that maps the graph of  on to the graph of

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

(i) On the diagram above, sketch the graph of and label the coordinates of the turning point.

(ii) State the vertical distance between the turning points on the graphs of and 

The diagram below shows the graph of .

The stationary points and -intercepts are marked on the diagram.

On separate diagrams, sketch the graphs with equations

(i)

(ii)

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

The graph  is transformed in the following order:

1. A translation by 

2. A reflection in the -axis

3. A vertical stretch by a scale factor of 

4. A translation by 

Find the equation of the graph after these transformations, in terms of .

Write in the form , where and are integers to be found.

Sketch the curve with equation showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Describe fully the transformation that maps the curve with equation onto the curve with equation where

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

(i)

(ii)

A function is given by

A different function, , is obtained by applying the following sequence of transformations to in order:

1. A translation by

2. A horizontal stretch by a scale factor of

3. A reflection in the -axis

Find an expression for , giving your answer in the form where , and are integers.

A sketch of the graph with equation , where

is shown below.

The points , and  are the points where the graph meets the coordinate axes.

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

On the graph of where and are non-zero constants, none of the points , and lie on the coordinate axes.

State the values that and cannot take.

The diagram shows the graph of , where

(i) Find the maximum value of  on the graph of

(ii) Find the value of  for which this maximum occurs.

Find in the form

where , , and are constants, a transformation that would map the graph of  onto the graph of

The diagram below shows the graph of  where

Find the equations of the asymptotes.

Find the equations of any asymptotes on the graph of

Find the range of the function where

The function is given by

where

•  is a non-zero constant

• the domain of is

The graph of is shown below.

Find the equations of the asymptotes on the graph of .

The function is defined by

• The domain of is

• The range of is

Find the constants and .

The point with coordinates  is the only stationary point on the curve with equation .

Find the coordinates of the stationary point on the following curves:

(i)

(ii)

(iii)

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

Sketch the graph of  and label the coordinates of the local maximum point.

Find the exact distance between the minimum point on the curve and the maximum point on the curve .

Figure 4 shows a sketch of the graph with equation

where is a positive constant.

Sketch the graph with equation where

stating

• the coordinates of the maximum point

• the coordinates of any points where the graph cuts the coordinate axes

Find, in terms of , the coordinates of the minimum point of the graph with equation

The diagram below shows the graph of .

The coordinates of the stationary points and intercepts with the axes are also shown, given in terms of where is a positive constant.

On separate diagrams, sketch the graphs with the following equations:

(i)

(ii)

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

Describe, in order, a sequence of transformations that maps the graph of   on to the graph of

where , and  are constants and .

Explain how the answer to part (a) changes in the cases when

(i)

(ii)

A function is given by

The curve  is transformed to the curve by applying the following transformations to in order:

1. A horizontal stretch of scale factor 3

2. The modulus of the function is then taken

3. A reflection in the -axis

Show that

where , and are constants to be found.

A function is given by

The graph of is transformed in the following order:

1. A horizontal stretch of scale factor , where

2. A translation of

3. A vertical stretch of scale factor

The equation of the graph after these transformations is , where is the inverse function of .

Find the values of the constants and .

A sketch of the graph with equation , where

with  is shown below.

The points , and  are points where the graph meets the coordinate axes.

Find, in terms of , the coordinates of , and .

Sketch the graph of

Label the coordinates of the images of the three points , and in terms of .

The graph of

is such that

• the images of the points , and lie on their original coordinate axes

• the image of is three times its previous distance from the origin

• the images of and are 20% of their previous distances from the origin

• the -coordinate of the point is negative

Find , , and .

The point with coordinates where is the only stationary point on the curve with equation .

Find, in terms of , the coordinates of the stationary point on the following curves:

(i)

(ii)

(iii)

A sketch of the graph with equation where

is shown below.

• The points and  are the -axis intercepts

• The point  is the maximum point

On the same diagram, sketch the graph of

Find the coordinates of the points , and , the images of the points , and respectively.

The area of triangle is times the area of triangle .

Find , giving a reason for your answer.

The diagram shows the graph of , where

• The vertical distance between the minimum point and the maximum point is 2

• The horizontal distance between the minimum point and the maximum point is 180

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

The exact distance between the minimum point and maximum point on the graph of

is given by

where is a positive integer.

Find the value of .