Transformations of Functions (Cambridge (CIE) AS Maths: Pure 1): Exam Questions

The point  lies on the curve with equation . 

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

(i)

(ii)

(iii)

(iv)

The point  lies on the curve with equation .  

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

(i)

(ii)

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 graph of The two marked points  and  lie on the graph.

In separate diagrams, sketch the curves with equation

(i)

(ii)

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

On the graph of  the image of one of the two marked points has an  coordinate of 2.  Find the two possible values of .

The diagram below shows the graph of .  The marked point  lies on the graph, and the graph meets the origin at the marked point .

In separate diagrams, sketch the curves with equation

(i)

(ii)

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

On the graph of  the image of one of the two marked points has a coordinate of 4. Find the value of .

The diagram below shows the graph of .  The graph intersects the coordinate axes at the two marked points  and .  The graph has two asymptotes as shown, with equations     and 

In separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, give the coordinates of the images of points  and under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of  has an asymptote at one of the coordinate axes.  Find the value of .

Sketch the graph of  , showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

The graph of   passes through the origin.  Find the value of .

Given that 

Sketch the graph of   showing clearly the coordinates of the points where the curve crosses the coordinate axes.

The graph with equation   passes through the point .  Find the three possible values of .

The point  lies on the curve with equation . 

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

(i)

(ii)

(iii)

(iv)

The point  lies on the curve with equation .

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

(i)

(ii)

The point  lies on the curve with equation

The graph is translated so that the point is mapped to the point .  Write down the equation of the transformed function.

The graph is translated so that the point is mapped to the point . 
Write down the equation of the transformed function in the form , where  is a constant to be found.

The point  lies on the curve with equation

The graph is stretched so that the point is mapped to the point .  Write down the equation of the transformed function in the form , where ,  and  are constants to be found.

The graph is stretched so that the point is mapped to the point . Write down the equation of the transformed function in the form , where  is a constant to be found.

The diagram below shows the graph of .  The two marked points  and  lie on the graph.

In separate diagrams, sketch the curves with equation

(i)

(ii)

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

On the graph of  the images of the two marked points both lie on the same side of the -axis.  Find the range of possible values of .

The diagram below shows the graph of . The marked point  lies on the graph, and the graph meets the origin at the marked point .

In separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, give the coordinates of the images of points  and  under the given 

The diagram below shows the graph of . The graph intersects the coordinate axes at the two marked points and .  The graph has two asymptotes as shown, with equations     and  .

In separate diagrams, sketch the curves with equation

(i)

(ii)

On each diagram, give the coordinates of the images of points  and  under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of  has an asymptote with equation .  Find the value of .

Sketch the graph of  , showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

The graph of    passes through the origin.  Find the two possible values of .

Given that 

Sketch the graph of  , showing clearly the coordinates of the points where the curve crosses the coordinate axes and the co-ordinates of any minimum points. (You do not need to state the co-ordinates of any maximum points).

The graph with equation     crosses the -axis three times.  Find the range of possible values of .

The curve with equation  has two asymptotes, for which the equations are

 and . 

Give the equations of the asymptotes for the curves with the following equations:

(i)

(ii)

The curve with equation has two asymptotes, for which the equations are and  

Give the equations of the asymptotes for the curves with the following equations:

(i)

(ii)

The curve with equation  has two asymptotes, for which the equations are  and . 

Give the equations of the asymptotes for the curves with the following equations:

(i)

(ii)

The diagram below shows the graph of .  The two marked points  and  lie on the graph.

In separate diagrams sketch the curves with equation

(i)

(ii)

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

On the graph of  the image of one of the two marked points has an  coordinate of .  Given that , find the value of .

The diagram below shows the graph of .  The marked point  lies on the graph, and the graph meets the origin at the marked point .

Consider the three following transformations of the graph

where  and  are constants, and

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

(i) The images of the two marked 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 graph of .  The graph intersects the coordinate axes at the two marked points and .  The graph has two asymptotes as shown, with equations      and     .

In separate diagrams sketch the curves with equation

(i)

(ii)

On each diagram give the coordinates of the images of points and B under the given transformation, as well as stating the equations of the transformed asymptotes.

The graph of  has an asymptote with equation , where .  Find the range of possible values of .

The function  is defined by the equation

Sketch the graph of , showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

The graph of   is such that, for all points  that lie on the graph, if the  coordinate is less than 5 then the  coordinate is less than zero.  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 equations

The graph of  touches the -axis at the point .  Find the value of , giving your answer as an exact value.