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

The diagram below shows the graph of .

The point  has coordinates , where

Giving your answers in terms of  and , find the coordinates of the image of the point  under the following graph transformations:

(i)

(ii)

(iii)

The function is defined by

Sketch the graph of .

On your sketch, show clearly

• the coordinates of the points where the graph intersects the coordinate axes

• the coordinates of the turning point

On separate diagrams, sketch the graphs of:

(i)

(ii)

In each case, show clearly

• the coordinates of the points where the graph intersects the coordinate axes

• the coordinates of the turning point

The point  lies on the curve with equation

(i) On the graph of , where is a constant, the point  is mapped to the point . Determine the value of  .

(ii) On the graph of , where is a constant, the point  is mapped to the point . Determine the value of .

(iii) On the graph of , where is a constant, the point is mapped to the point . Determine the value of .

(iv) On the graph of , where is a constant, the point is mapped to the point . Determine the value of .

The diagram below shows the curve with equation .

The points  and are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the stationary points.

On the graph of , where is a constant, the coordinate of one of the stationary points is 2. 

Find the possible values of .

The diagram below shows the curve with equation .

The points  and are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the stationary points.

On the graph of , where is a constant, the coordinate of one of the stationary points is 4.

Find the value of .

The diagram below shows the curve with equation .

The curve intersects the coordinate axes at the two points,  and . 

The curve has two asymptotes, as shown, with equations   and .

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the images of points  and under the given transformation and the equations of any asymptotes.

The curve with equation  has an asymptote along one of the coordinate axes. 

Find the value of .

Sketch the curve

Label clearly the coordinates of any points where the curve crosses the coordinate axes and give the equations of any asymptotes.

The curve with equation

passes through the origin. 

Find the value of .

Given that 

sketch the curve with equation  showing clearly the coordinates of the points where the curve crosses the coordinate axes.

The curve with equation 

passes through the point . 

Find the three possible values of .

The point  lies on the curve with equation . 

Find the coordinates of the image of the point  on the curves with the following equations:

(i)

(ii)

(iii)

(iv)

The point  lies on the curve with equation .

Find the coordinates of the image of the point  on the curves with the following equations:

(i)

(ii)

The diagram below shows the curve with equation .

The points  and are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the stationary points.

The diagram below shows the curve with equation .

The curve intersects the coordinate axes at the two points,  and . 

The curve has two asymptotes, as shown, with equations   and .

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the images of points  and under the given transformation and the equations of any asymptotes.

The graph of , where is a constant, has an asymptote with equation . 

Find the value of .

The curve with equation  has two asymptotes with equations  and .

Find the equations of the asymptotes for the following curves:

(i)

(ii)

The curve with equation  has two asymptotes with equations and  

Find the equations of the asymptotes for the following curves:

(i)

(ii)

The curve with equation  has two asymptotes with equations  and . 

Find the equations of the asymptotes for the following curves:

(i)

(ii)

The curve has equation where

Write in the form where is a quadratic expression to be found.

Hence deduce the coordinates of the points of intersection of the curve with equation

and the coordinate axes.

The point  lies on the curve with equation .

The curve is translated so that the point is mapped to the point . 

Find the equation of the transformed curve.

If, instead, the curve is translated so that the point is mapped to the point , find the equation of the transformed curve.

Give your answer in the form , where  is a constant to be found.

The point  lies on the curve with equation .

The curve is stretched so that the point is mapped to the point .

Find the equation of the transformed curve.

Give your answer in the form , where ,  and  are constants to be found.

If, instead, the curve is translated so that the point is mapped to the point , find the equation of the transformed curve.

Give your answer in the form , where  is a constant to be found.

The diagram below shows the curve with equation .

The points  and are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the stationary points.

On the graph of , the two stationary points both lie on the same side of the -axis. 

Find all the possible values that can take.

Sketch the curve with equation

Label clearly the coordinates of any points where the curve crosses the coordinate axes and give the equations of any asymptotes.

The curve with equation

passes through the origin. 

Find the two possible values of .

The diagram below shows the curve with equation .

The points  and are the stationary points on the curve.

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the stationary points.

On the graph of , where   is a constant, the  coordinate of one of the stationary points is .

Given that , find the value of .

The diagram below shows the curve with equation .

The points  and are the stationary points on the curve.

Consider the three following graph transformations

where  and  are constants and .

State which of the transformations satisfies each of the following conditions, and determine the range of possible values of  and  where necessary.

(i) The images of the stationary points under the transformation lie on opposite sides of the -axis.

(ii) The image of point under the transformation has coordinates , where .

(iii) The image of point  under the transformation has coordinates , where .

.

The diagram below shows the curve with equation .

The curve intersects the coordinate axes at the two points,  and . 

The curve has two asymptotes, as shown, with equations   and .

On separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, label the coordinates of the images of points  and under the given transformation and the equations of any asymptotes.

The graph of , where  is a constant, has an asymptote with equation , where . 

Find the range of possible values of .

Given that 

sketch the graph of , showing clearly the coordinates of the points where the curve crosses or touches the coordinate axes.

The functions  and  are defined by

The graph of  touches the -axis at the point . 

Find the exact value of .

The function  is defined by

Sketch the graph of , showing clearly

• the coordinates of the points where the curve crosses the coordinate axes

• the equations of any asymptotes

The graph of   is such that any point on the graph with a -coordinate of less than 5 has a negative -coordinate.

Find the range of possible values of .