Edexcel International A Level Maths: Pure 3

Topic Questions

1.3 Combinations of Transformations

1
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4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.

The stationary point A left parenthesis negative 2 space comma 8 right parenthesis is marked on the diagram.

q1-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-easy

On separate diagrams, sketch the following graphs

(i)

y equals 2 straight f left parenthesis x right parenthesis plus 1

(ii)
y equals begin inline style 1 half end style straight f left parenthesis x plus 1 right parenthesis

On each diagram, state the coordinates of the image of point A under the given transformation.

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2
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4 marks

Given that y equals straight f left parenthesis x right parenthesis, find equations, in terms of straight f left parenthesis x right parenthesis, for the following transformations:

(i)

Translation by open parentheses table row cell negative 2 end cell row 3 end table close parentheses

(ii)
Horizontal stretch of scale factor 2 followed by a vertical stretch of scale factor 3

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3a
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1 mark

The graph of y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis equals 2 x minus 3 is shown below.

q3a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-easy

Determine the coordinates of the points marked A and B.

3b
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5 marks
(i)

On the diagram above sketch the graph of y equals vertical line straight f left parenthesis x minus 1 right parenthesis vertical line.

(ii)
Determine the coordinates of the image of the points A spaceand B under the transformation in part (i).

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4
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3 marks

Describe, in order, a sequence of transformations of the graph of y equals straight f left parenthesis x right parenthesis given by the following equations:

(i)

y equals 3 straight f left parenthesis x right parenthesis minus 1

(ii)
y equals begin inline style 1 third end style straight f left parenthesis x minus 1 right parenthesis

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5a
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2 marks

The function straight g left parenthesis x right parenthesis is given as straight g left parenthesis x right parenthesis equals 2 x.

On the same diagram, sketch the graphs of y equals straight g left parenthesis x right parenthesis and y equals straight g to the power of negative 1 end exponent left parenthesis x right parenthesis
Label the coordinates of the points where each graph crosses the coordinate axes.

5b
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4 marks
(i)

Write down an expression for straight g to the power of negative 1 end exponent left parenthesis x right parenthesis in terms of x.

(ii)
Find an expression for straight g to the power of negative 1 end exponent left parenthesis x right parenthesis in terms of straight g left parenthesis x right parenthesis and state the type of transformation this would be.

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6
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4 marks

The equation y equals straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis squared, is shown below..

q6-2-10-cobinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-easy

The points A spaceand B are the points where the graph intersects the coordinate axes.

(i)

Write down the coordinates of A spaceand B.

(ii)
The graph of y equals straight f left parenthesis x right parenthesis is transformed to the graph of y equals negative straight f left parenthesis x right parenthesis minus 4.
Find the coordinates of the images of points A spaceand B under these transformations.

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7a
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2 marks

The diagram shows the graph of y equals straight f left parenthesis t right parenthesis, where straight f left parenthesis t right parenthesis equals cos space t0 degree less or equal than straight t less or equal than 360 degree.

q7a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-easy

(i)

Write down the minimum value of y equals straight f left parenthesis t right parenthesis in the given domain for straight t.

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

7b
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2 marks
(i)

Write down the minimum value of y equals 3 straight f left parenthesis straight t minus 45 degree right parenthesis in the given domain for t.

(ii)
Write down the value of t for which this minimum occurs.
7c
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2 marks

Find, in terms of straight f left parenthesis t right parenthesis, the combination of transformations that would map the graph of y equals straight f left parenthesis t right parenthesis onto the graph of y equals 3 space cos space t plus 10 degree less or equal than t less or equal than 360 degree.

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8
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3 marks

The function straight f left parenthesis x right parenthesis is to be transformed by a sequence of functions, in the order detailed below:

  1. A reflection in the y-axis.
  2. A vertical stretch of scale factor 4.
  3. A translation by open parentheses table row 0 row cell negative 2 end cell end table close parentheses

Write down an expression for the combined transformation in terms of straight f left parenthesis x right parenthesis.

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9
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4 marks

The diagram below shows the graph of y equals straight g left parenthesis x right parenthesis where

g left parenthesis x right parenthesis equals 1 over x comma space space space space space space space space space space space space space x not equal to 0

q9-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-easy

(i)

Write down the equations of the two asymptotes.

(ii)
Determine the equations of the two asymptotes on the graph of y equals straight g left parenthesis x minus 1 right parenthesis plus 5.

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1
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6 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis .
The stationary points and intercepts with the x-axis are marked on the diagram.

q1-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-hard

On separate diagrams, sketch the graphs with equations

(i)      y equals straight f left parenthesis 1 half space x right parenthesis plus 2 comma

(ii)

y equals negative straight f left parenthesis x minus 1 right parenthesis

On each diagram, mark the coordinates of the images of the points A comma B spaceand C under the given transformation.

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2
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4 marks

Given that straight f left parenthesis x right parenthesis equals ln left parenthesis 2 x plus 1 right parenthesis find an expression for straight g left parenthesis x right parenthesis, where straight g left parenthesis x right parenthesis is obtained by applying the following sequence of transformations to straight f left parenthesis x right parenthesis.

  1.    Translation by open parentheses table row cell negative 3 end cell row 0 end table close parentheses,
  2.    Horizontal stretch by scale factor begin inline style 1 half end style,
  3.    Reflection in the x-axis.

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3a
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3 marks

On the same axes sketch the graphs of y equals straight p left parenthesis x right parenthesis and y equals p to the power of negative 1 end exponent left parenthesis x right parenthesis, where straight p left parenthesis x right parenthesis equals vertical line 2 x vertical line comma space space x less or equal than 0.

3b
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3 marks

Find an expression for straight p to the power of negative 1 end exponent left parenthesis x right parenthesis and state its domain.

3c
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3 marks

Show that straight p to the power of negative 1 end exponent left parenthesis x right parenthesis equals negative 1 half straight p left parenthesis negative begin inline style 1 half end style x right parenthesis.

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4a
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3 marks

A sketch of the graph with equation y equals straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis equals left parenthesis x squared minus 4 right parenthesis squared is shown below.

q6a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-hard

The points A comma B spaceand C are the points where the graph intercepts the coordinate axes.

Sketch the graph of y equals negative 3 straight f left parenthesis 2 x right parenthesis, labelling the images of the three points A comma B spaceand C.

4b
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2 marks

Suggest a combination of at least two transformations that will transform the points A comma B spaceand C such that none of them lie on the coordinate axes.

Give your answer in the form of an expression in terms of straight f left parenthesis x right parenthesis.

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5a
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2 marks

The diagram shows the graph of y equals straight f left parenthesis t right parenthesis, where straight f left parenthesis t right parenthesis equals cos space t,  0 degree less or equal than x less or equal than 360 degree.

q7a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-hard

(i)

Write down the maximum value of y when y equals negative 2 straight f left parenthesis 3 t right parenthesis.

(ii)
Write down the value of t for which this maximum occurs.
5b
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2 marks

Find, in terms of straight f left parenthesis t right parenthesis, the combination of transformations that would map the graph of y equals straight f left parenthesis t right parenthesis onto the graph of y equals 2 minus 4 sin space t,0 degree less or equal than x less or equal than 180 degree.

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6
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3 marks

The function straight f left parenthesis x right parenthesis is to be transformed by a sequence of functions, in the order detailed below.

  1. A translation by open parentheses table row 2 row 0 end table close parentheses
  2. A reflection in the y-axis
  3. A vertical stretch by scale factor 2 over 3
  4. A translation by open parentheses table row 0 row 4 end table close parentheses

Write down the combined transformation in terms of straight f left parenthesis x right parenthesis.

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7a
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3 marks

The minimum point on the graph of y equals straight f left parenthesis x right parenthesis has coordinates left parenthesis 4 space comma negative 8 right parenthesis as shown on the diagram below.

q10a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-hard

Sketch the graph of y equals vertical line straight f left parenthesis 2 x right parenthesis vertical line minus 3 and state the coordinates of the maximum point.

7b
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2 marks

Find the exact distance between the minimum point on the graph of y equals straight f left parenthesis x right parenthesis and the maximum point on the graph of y equals vertical line straight f left parenthesis 2 x right parenthesis vertical line minus 3.

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1
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4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.
The stationary points are marked on the diagram.

q1-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-medium

On separate diagrams, sketch the graphs with equation

(i)

y equals 2 straight f left parenthesis x right parenthesis minus 4

(ii)
y equals straight f left parenthesis x plus 1 right parenthesis plus 3

On each diagram, state the coordinates of the images of the points A spaceand B under the given transformation.

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2
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3 marks

Describe, in order, a sequence of transformations that maps the graph of y space equals space straight f left parenthesis x right parenthesis onto the following graphs:

(i)
y space equals space 3 straight f left parenthesis x plus 2 right parenthesis,
(ii)
y space equals space straight f left parenthesis negative x right parenthesis minus 1.

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3
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4 marks

Given that straight f left parenthesis x right parenthesis space equals space 3 x squared minus 2 x find an expression for g left parenthesis x right parenthesis, where g left parenthesis x right parenthesis is obtained by applying the following sequence of transformations to straight f left parenthesis x right parenthesis.

   1.    Translation by open parentheses table row 2 row 0 end table close parentheses

   2.   Vertical stretch of scale factor 4

   3.   Translation by open parentheses table row 0 row cell negative 3 end cell end table close parentheses

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4a
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4 marks
(i)
Sketch the graph of y equals straight p left parenthesis x right parenthesis, where straight p left parenthesis x right parenthesis equals 3 x minus 4.

(ii)
On the same set of axes, sketch the graph of y equals p to the power of negative 1 end exponent left parenthesis x right parenthesis.
Label the coordinates of the points where each graph crosses the coordinate axes   
      
4b
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4 marks
(i)

      Find an expression for straight p to the power of negative 1 end exponent left parenthesis x right parenthesis.

(ii)     Find an expression for 1 over 9 space left square bracket straight p left parenthesis x right parenthesis plus 16 right square bracket.

(iii)
What can you deduce about the sequence of transformations given
by begin inline style 1 over 9 end style left square bracket straight p left parenthesis x right parenthesis plus 16 right square bracket ?

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5a
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2 marks

The equation y equals straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis equals left parenthesis x minus a right parenthesis squared, with a greater than 1, is shown below.

q6a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-medium

The points A spaceand B are the points where the graph intercepts the coordinate axes.

Write down, in terms of a, the coordinates of A spaceand B.

5b
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3 marks

Sketch the graph of y equals negative straight f left parenthesis negative x right parenthesis, labelling the images of the points A spaceand B stating their coordinates in terms of a.

5c
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1 mark

Write down the value of  such that the point A is three times as far from the origin as the point B.

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6a
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2 marks

The diagram shows the graph of y equals straight f left parenthesis t right parenthesis, where straight f left parenthesis t right parenthesis equals sin space 2 t, 0 degree less or equal than x less or equal than 180 degree.

q7a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-medium

(i)

Write down the maximum value of y when y equals 3 straight f left parenthesis t right parenthesis.

(ii)
Write down the first value of t for which this maximum occurs.
6b
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2 marks
(i)

Write down the minimum value of  when y equals 5 straight f left parenthesis t plus 30 degree right parenthesis.

(ii)

Write down the first value of t for which this minimum occurs.

6c
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2 marks

Find, in terms of straight f left parenthesis t right parenthesis, the combination of transformations that would map the graph of y equals straight f left parenthesis t right parenthesis onto the graph of y equals 2 plus sin space t0 degree less or equal than x less or equal than 180 degree .

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7
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3 marks

The function straight f left parenthesis x right parenthesis 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 x-axis
  3. A translation by open parentheses table row 0 row 2 end table close parentheses

Write down an expression for the combined transformation in terms of straight f left parenthesis x right parenthesis.

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8a
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2 marks

The diagram below shows the graph of y equals straight g left parenthesis x right parenthesis where

straight g left parenthesis x right parenthesis equals fraction numerator 2 x plus 1 over denominator x minus 1 end fraction comma space space space space space space space space space space space space x not equal to 1

q8a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-medium

Write down the equations of the two asymptotes.

8b
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2 marks

Determine the equations of the two asymptotes on the graph of y equals straight g left parenthesis 2 x right parenthesis minus 3.

8c
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2 marks

Determine the range of vertical line straight g left parenthesis 3 x right parenthesis minus 2 vertical line.

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9
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3 marks

The point with coordinates left parenthesis 1 space comma negative 4 right parenthesis is a stationary point on the graph with equation y equals straight h left parenthesis x right parenthesis.

Determine the coordinates of the stationary point on the graphs with the following equations:

(i)

y equals 2 straight h left parenthesis x minus 1 right parenthesis,

(ii)

y equals negative straight h left parenthesis x plus 1 right parenthesis plus 2,

(iii)
y equals vertical line straight h left parenthesis 3 x right parenthesis plus 2 vertical line.

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10
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4 marks

The turning point on the graph of y equals straight f left parenthesis x right parenthesis has coordinates left parenthesis 2 space comma negative 5 right parenthesis as shown on the diagram below.

q12-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-medium

(i)

On the diagram above sketch the graph of y equals vertical line straight f left parenthesis x right parenthesis vertical line plus 1  and state the coordinates of
the turning point.

(ii)
State the distance between the turning points on the graphs of y equals straight f left parenthesis x right parenthesis and y equals vertical line straight f left parenthesis x right parenthesis vertical line plus 1.
.

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1a
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4 marks

Describe, in order, a sequence of transformations that would map the graph of y equals straight f left parenthesis x right parenthesis
 onto each of the following graphs:

(i)

y equals a straight f left parenthesis x plus b right parenthesis plus c  for the case when a greater than 0.

(ii)
y equals negative straight f left parenthesis negative x right parenthesis
1b
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2 marks

How, if at all, would your answer to part (a) (i) change if a equals 1 or if a less than 0?

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2
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4 marks

The function straight f left parenthesis x right parenthesis equals e to the power of 3 x end exponent minus x minus 6 is transformed by a sequence of transformations as described below.

    1.    Horizontal stretch by scale factor 3,
    2.    The modulus of the function is then taken,
    3.    Reflection in the y-axis.

Write down the resulting transformation in terms of straight f left parenthesis x right parenthesis as well as an expression in terms of x.

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3
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4 marks

Show that the graph of y equals straight p left parenthesis x right parenthesis where straight p left parenthesis x right parenthesis equals 2 x plus 1 maps onto the graph of its inverse under the transformations described by 1 half straight p space left parenthesis 1 half space x right parenthesis minus 1.

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4
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4 marks

Prove, that for a constant k comma space k not equal to 0, if straight f left parenthesis x right parenthesis equals k x, then straight f to the power of negative 1 end exponent left parenthesis x right parenthesis equals 1 over k straight f space left parenthesis 1 over k x right parenthesis.

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5a
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2 marks

A sketch of the graph with equation y equals straight f left parenthesis x right parenthesis, where straight f left parenthesis x right parenthesis equals left parenthesis x squared minus a right parenthesis squared, with a greater than 1 is shown below.

q7a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-veryhard

The points A comma B spaceand C are points where the graph intercepts the coordinate axes.

Write down, in terms of a, the coordinates of A comma B spaceand C.

5b
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3 marks

Sketch the graph of y equals negative fraction numerator 1 space over denominator 2 end fraction straight f left parenthesis x minus 1 right parenthesis, labelling the images of the three points A comma B spaceand Cand stating their coordinates in terms of a.

5c
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2 marks

Suggest, in terms of straight f left parenthesis x right parenthesis, a combination of at least two transformations, such that the points A comma B spaceand C spacetransform to new positions but remain lying on their respective axes.

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6a
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4 marks

The diagram shows the graph of y equals straight f left parenthesis t right parenthesis, where straight f left parenthesis t right parenthesis equals cosec space t0 degree less or equal than x less or equal than 360 degree.

q8a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-veryhard

The vertical distance between the minimum point, left parenthesis 90 space comma 1 right parenthesis , and the maximum point, left parenthesis 270 space comma negative 1 right parenthesis is 2.  The horizontal distance between them is 180.

Find, in terms of a, the vertical and horizontal distances between the minimum and maximum point on the graph of y equals negative 1 over a space straight f left parenthesis a t right parenthesis comma space a not equal to 0.

6b
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2 marks

Hence or otherwise show that the distance between the minimum and maximum point on the graph of y equals negative begin inline style 1 over a end style space straight f left parenthesis a t right parenthesis comma space a not equal to 0, i                     

                     fraction numerator 2 square root of 8101 over denominator a end fraction

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7a
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3 marks

A sketch of the graph with equation y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis equals 10 x minus x squared minus 16 is shown below.

Points A spaceand B are the x-axis intercepts and point C is the maximum point on the graph.

q9a-2-10-combinations-of-transformations-a-level-only-edexcel-a-level-pure-maths-veryhard

On the diagram above, sketch the graph of y equals negative vertical line 1 fourth space straight f left parenthesis 1 half space x right parenthesis vertical linelabelling the image of the points A comma B and C with A to the power of apostrophe comma B to the power of apostrophe spaceand C apostrophe.

7b
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4 marks

Show that the area of A B C is twice the area of triangle A apostrophe B apostrophe C apostrophe.

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