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A LevelMaths: PureAQAExam Questions2. Algebra & Functions2.9 Transformations of Functions

Transformations of Functions (AQA A Level Maths: Pure)

Exam Questions

3 hours32 questions
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1
Sme Calculator4 marks

A curve has equation y equals straight f open parentheses x close parentheses.

Describe the transformation of the curve given by the equations below:

(i)

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

(ii)

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

(iii)

straight y equals 3 straight f left parenthesis x right parenthesis comma

(iv)
y equals straight f left parenthesis 2 x right parenthesis.

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2
Sme Calculator4 marks

A curve has equation y equals straight f open parentheses x close parentheses.

Write down the equations of the curves, in terms of straight fopen parentheses x close parentheses, given by the following transformations:

(i)

Translation by the vector open parentheses table row 3 row 0 end table close parentheses,

(ii)

Horizontal stretch, scale factor 2,

(iii)

Vertical stretch, scale factor 1 third,

(iv)
Reflection in the y-axis.

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3
Sme Calculator3 marks

The point P open parentheses 2 comma 6 close parentheses lies on the curve with equation y equals f left parenthesis x right parenthesis.

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

(i)

y equals straight f open parentheses x close parentheses plus 1

(ii)

y equals negative straight f open parentheses x close parentheses

(iii)
y equals straight f left parenthesis 1 fourth x right parenthesis.

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4
Sme Calculator3 marks

A point P open parentheses negative 2 comma 8 close parentheses, on the graph of y equals straight f open parentheses x close parentheses, is mapped to the point P to the power of apostrophe under a single transformation.

For the following coordinates of P to the power of apostrophe write down what the transformation could have been:

(i)

P to the power of apostrophe left parenthesis negative 2 comma 3 right parenthesis comma

(ii)

P to the power of apostrophe left parenthesis negative 4 comma 8 right parenthesis comma

(iii)
P to the power of apostrophe left parenthesis negative 2 comma negative 8 right parenthesis.

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5
Sme Calculator3 marks

Point P has coordinates left parenthesis 3 space comma negative 4 right parenthesis space and lies on the curve with equation space y equals straight f left parenthesis x right parenthesis..

Write down the value of a given that:

(i)

On the graph of y equals straight f open parentheses x italic plus straight a close parentheses, point P is mapped to point P to the power of apostrophe left parenthesis negative 3 comma negative 4 right parenthesis comma

(ii)

On the graph of y equals a straight f left parenthesis x right parenthesis comma space, point P is mapped to point P to the power of apostrophe left parenthesis 3 comma negative 12 right parenthesis comma

(iii)
On the graph of y equals straight f left parenthesis a x right parenthesis comma space, point P is mapped to point space P apostrophe left parenthesis negative 3 space comma negative 4 right parenthesis.

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6a
Sme Calculator3 marks

The function straight f left parenthesis x right parenthesis is defined as straight f left parenthesis x right parenthesis equals left parenthesis x minus 2 right parenthesis left parenthesis x minus 6 right parenthesis

Sketch the graph of y equals straight f left parenthesis x right parenthesis comma space, showing clearly the coordinates of the points where the graph intersects the coordinate axes and the coordinates of the turning point.

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6b
Sme Calculator4 marks

On separate diagrams sketch the graphs of:

(i)

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

(ii)
y equals straight f left parenthesis negative x right parenthesis.

In each case clearly show the coordinates of the points where the graph intersects the coordinate axes and the coordinates of the turning point.

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7
Sme Calculator3 marks

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

The point P has coordinates open parentheses a comma b close parentheses, where a comma b greater than 0.

q7-2-9-transformations-of-functions-edexcel-a-level-pure-maths-easy

In terms of a and b write down the coordinates of the image of point P under the following graph transformations:

(i)

y equals straight f left parenthesis 2 x right parenthesis comma

(ii)

y equals negative straight f left parenthesis x right parenthesis

(iii)
y equals a straight f left parenthesis x right parenthesis

.

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1
Sme Calculator4 marks

The point P left parenthesis negative 1 comma 4 right parenthesis space lies on the curve with equation space y equals straight f left parenthesis x right parenthesis. space. 

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

(i)

y equals straight f left parenthesis x right parenthesis plus 3

(ii)

y equals straight f left parenthesis x plus 3 right parenthesis

(iii)

y equals 3 straight f left parenthesis x right parenthesis

(iv)
y equals straight f left parenthesis 3 x right parenthesis space

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2
Sme Calculator2 marks

The point P left parenthesis negative 3 comma negative 4 right parenthesis lies on the curve with equation y equals straight f left parenthesis x right parenthesis. space.  

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

(i)
y equals straight f left parenthesis negative x right parenthesis

(ii)
space y equals negative straight f left parenthesis x right parenthesis

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3
Sme Calculator4 marks

The point P left parenthesis 3 comma 2 right parenthesis space lies on the curve with equation space y equals straight f left parenthesis x right parenthesis.

(i)
On the graph of y equals straight f left parenthesis x right parenthesis plus a, where a spaceis a constant, the point P is mapped to the point left parenthesis 3 comma negative 5 right parenthesis.  Determine the value of  a.
(ii)
On the graph of y equals straight f left parenthesis x plus b right parenthesis, where b spaceis a constant, the point P is mapped to the point left parenthesis negative 1 comma 2 right parenthesis. space  Determine the value of b.
(iii)
On the graph of space y equals c straight f left parenthesis x right parenthesis comma, where c spaceis a constant, the point P spaceis mapped to the point space left parenthesis 3 comma 1 right parenthesis.  Determine the value of c.
(iv)
On the graph of y equals straight f left parenthesis d x right parenthesis comma, where d spaceis a constant, the point P is mapped to the point left parenthesis 1 comma 2 right parenthesis.  Determine the value of d.

 

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4a
Sme Calculator4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis. space spaceThe two marked points A left parenthesis negative 1 comma 5 right parenthesis and B left parenthesis 3 comma negative 3 right parenthesis lie on the graph.

q4-2-9-transformations-of-functions-edexcel-a-level-pure-maths-medium

In separate diagrams, sketch the curves with equation

(i)

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

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

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

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4b
Sme Calculator2 marks

On the graph of y equals straight f left parenthesis x plus a right parenthesis the image of one of the two marked points has an  x coordinate of 2.  Find the two possible values of a.

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5a
Sme Calculator4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis..  The marked point B left parenthesis 4 comma 8 right parenthesis lies on the graph, and the graph meets the origin at the marked point A.

q5a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-medium

In separate diagrams, sketch the curves with equation

(i)

y equals negative straight f left parenthesis x right parenthesis

(ii)
y equals straight f left parenthesis 4 x right parenthesis

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

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5b
Sme Calculator2 marks

On the graph of y equals a straight f left parenthesis x right parenthesis the image of one of the two marked points has a y spacecoordinate of 4. Find the value of a.

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6a
Sme Calculator6 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.  The graph intersects the coordinate axes at the two marked points A left parenthesis 0 comma 6 right parenthesis and B left parenthesis 3 comma 0 right parenthesis.  The graph has two asymptotes as shown, with equations  space y equals fraction numerator space 20 over denominator 3 end fraction   and  x equals 10 over 3

q6a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-medium

In separate diagrams, sketch the curves with equation

(i)
y equals straight f left parenthesis x right parenthesis minus 6

(ii)
space y equals straight f left parenthesis negative x right parenthesis

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

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6b
Sme Calculator2 marks

The graph of y equals straight f left parenthesis x plus a right parenthesis has an asymptote at one of the coordinate axes.  Find the value of a.

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7a
Sme Calculator4 marks

Sketch the graph of  y equals 1 over x plus 3, showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

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7b
Sme Calculator1 mark

The graph of y equals fraction numerator space 1 over denominator open parentheses x plus a close parentheses end fraction plus 3 space  passes through the origin.  Find the value of a.

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8a
Sme Calculator4 marks

Given that x cubed minus 10 x squared minus 24 x equals x left parenthesis x plus 2 right parenthesis left parenthesis x minus 12 right parenthesis

Sketch the graph of  y equals x cubed minus 10 x squared minus 24 x comma showing clearly the coordinates of the points where the curve crosses the coordinate axes.

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8b
Sme Calculator2 marks

The graph with equation y equals left parenthesis x plus a right parenthesis cubed minus 10 left parenthesis x plus a right parenthesis squared minus 24 left parenthesis x plus a right parenthesis  passes through the point space left parenthesis negative 2 comma 0 right parenthesis.  Find the three possible values of a.

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1
Sme Calculator4 marks

The point space P left parenthesis negative 3 comma negative 2 right parenthesis lies on the curve with equation y equals straight f left parenthesis x right parenthesis. 

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

(i)

y minus 2 equals straight f left parenthesis x right parenthesis minus 6

(ii)

y equals straight f left parenthesis x minus 3 right parenthesis

(iii)

2 y equals straight f left parenthesis x right parenthesis

(iv)
y equals straight f left parenthesis 1 half space x right parenthesis

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2
Sme Calculator2 marks

The point P left parenthesis 0 comma 5 right parenthesis lies on the curve with equation y equals straight f left parenthesis x right parenthesis.

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

(i)

y equals straight f left parenthesis negative x right parenthesis

(ii)
negative y equals straight f left parenthesis x right parenthesis

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3a
Sme Calculator2 marks

The point P left parenthesis negative 12 comma negative 9 right parenthesis lies on the curve with equation y equals x squared plus 15 x plus 27.

The graph is translated so that the point P spaceis mapped to the point left parenthesis negative 12 comma 3 right parenthesis.  Write down the equation of the transformed function.

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3b
Sme Calculator2 marks

The graph is translated so that the point P spaceis mapped to the point left parenthesis negative 10 comma negative 9 right parenthesis. 
Write down the equation of the transformed function in the form y equals left parenthesis x plus a right parenthesis squared plus 15 left parenthesis x plus a right parenthesis plus 27, where a is a constant to be found.

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4a
Sme Calculator2 marks

The point P left parenthesis 3 comma negative 12 right parenthesis lies on the curve with equation y equals x squared minus 12 x plus 15.

The graph is stretched so that the point P spaceis mapped to the point left parenthesis 3 comma negative 4 right parenthesis. space space.  Write down the equation of the transformed function in the form y equals a x squared plus b x plus c, where a,b  and c are constants to be found.

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4b
Sme Calculator2 marks

The graph is stretched so that the point P spaceis mapped to the point left parenthesis 1 comma negative 12 right parenthesis. Write down the equation of the transformed function in the form y equals left parenthesis d x right parenthesis squared minus 12 left parenthesis d x right parenthesis plus 15, where d is a constant to be found.

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5a
Sme Calculator4 marks

The diagram below shows the graph of y equals f left parenthesis x right parenthesis.  The two marked points A left parenthesis negative 1 comma 5 right parenthesis and B left parenthesis 3 comma negative 3 right parenthesis lie on the graph.

q5a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-hard

In separate diagrams, sketch the curves with equation

(i)

y equals straight f left parenthesis negative x right parenthesis

(ii)
negative y equals straight f left parenthesis x right parenthesis

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

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5b
Sme Calculator3 marks

On the graph of y equals straight f left parenthesis x plus a right parenthesis the images of the two marked points both lie on the same side of the y-axis.  Find the range of possible values of a.

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6
Sme Calculator4 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis. The marked point B left parenthesis 4 comma 8 right parenthesis lies on the graph, and the graph meets the origin at the marked point A.

q6-2-9-transformations-of-functions-edexcel-a-level-pure-maths-hard

In separate diagrams, sketch the curves with equation

(i)

y equals straight f left parenthesis 1 third space x right parenthesis

(ii)
6 y equals straight f left parenthesis x right parenthesis

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

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7a
Sme Calculator6 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis. The graph intersects the coordinate axes at the two marked points A left parenthesis 0 comma 6 right parenthesis and B left parenthesis 3 comma 0 right parenthesis.  The graph has two asymptotes as shown, with equations  y equals fraction numerator space 20 over denominator 3 end fraction   and  x equals fraction numerator space 10 over denominator 3 end fraction .

q7a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-hard

In separate diagrams, sketch the curves with equation

(i)

y equals straight f left parenthesis 5 x right parenthesis

(ii)
space y equals negative straight f left parenthesis x right parenthesis

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

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7b
Sme Calculator1 mark

The graph of y equals a straight f left parenthesis x right parenthesis has an asymptote with equation y equals 2.  Find the value of a.

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8a
Sme Calculator4 marks

Sketch the graph of  y equals 2 minus 8 over x squared, showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

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8b
Sme Calculator2 marks

The graph of  y equals 2 minus 8 over open parentheses x plus a close parentheses squared  passes through the origin.  Find the two possible values of a.

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9a
Sme Calculator4 marks

Given that x cubed minus 8 x squared plus 16 x equals x left parenthesis x minus 4 right parenthesis squared

Sketch the graph of y equals x cubed minus 8 x squared plus 16 x plus 3 , 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).

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9b
Sme Calculator2 marks

The graph with equation   y plus a equals x cubed minus 8 x squared plus 16 x  crosses the x-axis three times.  Find the range of possible values of a.

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1
Sme Calculator4 marks

The curve with equation space y equals straight f left parenthesis x right parenthesis has two asymptotes, for which the equations are

y equals negative 3 and x equals 2. 

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

(i)

y plus 3 equals straight f left parenthesis x right parenthesis

(ii)
y equals straight f left parenthesis x minus 2 right parenthesis

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2
Sme Calculator4 marks

The curve with equation y equals straight f left parenthesis x right parenthesis has two asymptotes, for which the equations are y equals 5 spaceand x equals negative 4. 

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

(i)

1 third space y equals straight f left parenthesis x right parenthesis

(ii)
y equals straight f left parenthesis 1 third x right parenthesis space

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3
Sme Calculator4 marks

The curve with equation y equals straight f left parenthesis x right parenthesis has two asymptotes, for which the equations are y equals negative 1 and x equals negative 2. 

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

(i)

space y equals straight f left parenthesis negative x right parenthesis

(ii)
space minus y equals straight f left parenthesis x right parenthesis

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4a
Sme Calculator4 marks

The diagram below shows the graph of space y equals straight f left parenthesis x right parenthesis.  The two marked points A left parenthesis negative 1 comma 5 right parenthesis and B left parenthesis 3 comma negative 3 right parenthesis lie on the graph.

q4a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-veryhard

In separate diagrams sketch the curves with equation

(i)

y equals straight f left parenthesis begin inline style 1 third end style space x right parenthesis

(ii)

space 5 y equals straight f left parenthesis x right parenthesis

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

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4b
Sme Calculator2 marks

On the graph of y equals straight f left parenthesis a x right parenthesis the image of one of the two marked points has an x coordinate of begin inline style 5 over 3 end style.  Given that a greater than 0, find the value of a.

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5
Sme Calculator5 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.  The marked point space B left parenthesis 4 comma 8 right parenthesis lies on the graph, and the graph meets the origin at the marked point A.

q5-2-9-transformations-of-functions-edexcel-a-level-pure-maths-veryhard

Consider the three following transformations of the graph

y equals straight f left parenthesis negative x right parenthesis space space space space space space space space space space space space space space y equals straight f left parenthesis a x right parenthesis space space space space space space space space space space space space space y equals straight f left parenthesis x right parenthesis plus b

where a and b are constants, and a greater than 0

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

(i)

The images of the two marked points under the transformation lie on opposite sides of the x-axis.

(ii)

The image of point B under the transformation has coordinates left parenthesis x comma y right parenthesis, where negative 6 less than x less than negative 3.

(iii)
The image of point B under the transformation has coordinates left parenthesis x comma y right parenthesis, where 0 less than x less than 3.

.

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6a
Sme Calculator6 marks

The diagram below shows the graph of y equals straight f left parenthesis x right parenthesis.  The graph intersects the coordinate axes at the two marked points A left parenthesis 0 comma 6 right parenthesis and B left parenthesis 3 comma 0 right parenthesis.  The graph has two asymptotes as shown, with equations   y equals fraction numerator space 20 over denominator 3 end fraction   and   x equals fraction numerator space 10 over denominator 3 end fraction  .

q6a-2-9-transformations-of-functions-edexcel-a-level-pure-maths-veryhard

In separate diagrams sketch the curves with equation

(i)

y equals straight f left parenthesis begin inline style 20 over 3 end style x right parenthesis

(ii)

space 5 y equals 4 straight f left parenthesis x right parenthesis

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

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6b
Sme Calculator2 marks

The graph of y equals straight f left parenthesis a x right parenthesis has an asymptote with equation x equals k, where 1 less than k less than 100.  Find the range of possible values of space a.

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7a
Sme Calculator6 marks

The function straight f left parenthesis x right parenthesis is defined by the equation

straight f left parenthesis x right parenthesis equals 9 minus 16 over open parentheses x minus 2 close parentheses squared

Sketch the graph of y equals straight f left parenthesis x right parenthesis, showing clearly the points where the curve crosses the coordinate axes and stating the equations of the asymptotes.

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7b
Sme Calculator2 marks

The graph of y equals straight f left parenthesis x plus a right parenthesis  is such that, for all points P left parenthesis x comma y right parenthesis that lie on the graph, if the y coordinate is less than 5 then the x coordinate is less than zero.  Find the range of possible values of a.

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8a
Sme Calculator6 marks

Given that straight f left parenthesis x right parenthesis equals x cubed minus left parenthesis 2 square root of 3 right parenthesis x squared plus 3 x

Sketch the graph of y equals straight f left parenthesis x right parenthesis , showing clearly the coordinates of the points where the curve crosses or touches the coordinate axes.

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8b
Sme Calculator3 marks

The functions straight g left parenthesis x right parenthesis and straight h left parenthesis x right parenthesis are defined by the equations

straight g left parenthesis x right parenthesis equals straight f left parenthesis negative x right parenthesis space straight h left parenthesis x right parenthesis equals straight g left parenthesis x plus a right parenthesis

The graph of straight h left parenthesis x right parenthesis touches the x-axis at the point left parenthesis 5 comma 0 right parenthesis.  Find the value of a, giving your answer as an exact value.

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