A LevelMathsEdexcelPureExam QuestionsAlgebra & FunctionsGraphs of Functions

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

Exam code: 9MA0

2 hours33 questions
Easy
Medium
Hard
Very Hard
1a
2 marks

Factorise completely 9 x minus x cubed

How did you do?

1b
2 marks

The curve C has equation

y equals 9 x minus x cubed

Sketch C showing the coordinates of the points at which the curve cuts the x-axis.

How did you do?

2a
2 marks

(i) Write down the value of x squared plus 3 x minus 4 when x equals 0.

(ii) Factorise x squared plus 3 x minus 4 

How did you do?

2b
3 marks

Sketch the curve with equation

space y equals x squared plus 3 x minus 4

showing the coordinates of the points at which the curve cuts the coordinate axes.

How did you do?

3
2 marks

Sketch the curve with equation

space y equals x squared minus 1

showing the coordinates of the points at which the curve cuts the coordinate axes.

How did you do?

4
2 marks

Sketch the curve with equation

space y equals 1 over x

stating the equations of any asymptotes.

How did you do?

5
2 marks

Use a graphical method to solve the simultaneous equations

y equals x

y equals negative x plus 2

You may use the coordinate axes below.

2-7-edexcel-alevel-maths-pure-q4easy

How did you do?

6
3 marks

By sketching the curves y equals x cubed and y equals 1 over x on the same coordinate axes, show that there are two real solutions to the equation

x cubed equals 1 over x

How did you do?

7a
2 marks

On the axes below, sketch the lines y equals 3 x minus 2 and y equals x plus 4.

2-7-edexcel-alevel-maths-pure-q7medium

How did you do?

7b
1 mark

Hence use a graphical method to solve the simultaneous equations

3 x minus y equals 2

x minus y equals negative 4

How did you do?

8a
1 mark

The diagram below shows the graph of  y equals a over x, where a greater than 0.

2-7-edexcel-alevel-maths-pure-q8medium

Sketch the graph of y equals a over x, where a less than 0.

How did you do?

8b
2 marks

In each case below, find the equation of the curve in the form y equals a over x that passes through the point given.

(i) left parenthesis negative 4 comma negative 5 right parenthesis 

(ii) left parenthesis negative 0.02 comma space 250 right parenthesis

How did you do?

1a
2 marks

Sketch the curve with equation

y equals k over x space space space space space space space space space space space space space x not equal to 0

where k is a positive constant.

How did you do?

1b
3 marks

Hence or otherwise, solve

16 over x less or equal than 2

How did you do?

2a
2 marks

Express 

2 x cubed plus 2 x squared minus 12 x

in the form

a x left parenthesis x plus b right parenthesis left parenthesis x plus c right parenthesis

where a, b and c are integers to be found.

How did you do?

2b
3 marks

Hence sketch the curve with equation

space y equals 2 x cubed plus 2 x squared minus 12 x

showing the coordinates of the points at which the curve cuts the coordinate axes.

How did you do?

3a
1 mark

Use the factor theorem to show that left parenthesis x minus 2 right parenthesis is a factor of the function

straight f left parenthesis x right parenthesis equals x cubed minus 2 x squared minus 4 x plus 8

How did you do?

3b
2 marks

Hence express straight f left parenthesis x right parenthesis as a product of three linear factors.

How did you do?

3c
3 marks

Sketch the curve y equals straight f left parenthesis x right parenthesis showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

4
2 marks

Sketch the curve with equation

y equals left parenthesis x plus 3 right parenthesis cubed

showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

5
2 marks

Sketch the curve with equation

y equals 1 over x to the power of n

in the case where n equals 2, stating the equations of any asymptotes.

How did you do?

6
3 marks

You are given that y is inversely proportional to x.

When x equals 3, y equals 12.

(i) Find an equation for y in terms of x.

(ii) Sketch the graph of y against x, where x greater than 0.

How did you do?

7
3 marks

Sketch the curve with equation 

y equals 2 x squared left parenthesis x plus 3 right parenthesis

showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

8a
3 marks

On the same diagram, sketch the curves with equationsy equals x left parenthesis x plus 2 right parenthesis left parenthesis x minus 1 right parenthesis and y equals 1 over x.

How did you do?

8b
1 mark

Hence find the number of real solutions to the equation

x left parenthesis x plus 2 right parenthesis left parenthesis x minus 1 right parenthesis space equals 1 over x

How did you do?

9a
1 mark

Show that left parenthesis x plus 2 right parenthesis is a factor of 2 x cubed minus 3 x squared minus 11 x plus 6.

How did you do?

9b
2 marks

Hence express 2 x cubed minus 3 x squared minus 11 x plus 6 in the form

open parentheses x plus 2 close parentheses open parentheses x plus a close parentheses open parentheses b x plus c close parentheses

where a, b and c are integers to be found.

How did you do?

9c
3 marks

Sketch the curve with equation

y equals 2 x cubed minus 3 x squared minus 11 x plus 6

showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

10
2 marks

Sketch the curve with equation

space y equals negative 2 over x squared

stating the equations of any asymptotes.

How did you do?

11a
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3 marks

On the same axes below, plot the curve y equals x squared plus 2 x minus 3 and the line y equals x minus 1.

2-7-edexcel-alevel-maths-pure-q3hard

How did you do?

11b
2 marks

Hence use a graphical method to solve the simultaneous equations

space y equals left parenthesis x plus 3 right parenthesis left parenthesis x minus 1 right parenthesis

x minus y equals 1

How did you do?

12
3 marks

You are given that y is inversely proportional to x.

When x equals 2, y equals 10.

(i) Find an equation for y in terms of x.

(ii) Sketch the graph of y against x, where x greater than 0.

How did you do?

13a
1 mark

The diagram below shows a sketch of the curve y equals a over x squared, where a greater than 0.

2-7-edexcel-alevel-maths-pure-q8hard

Sketch the graph of y equals a over x squared, where a less than 0.

How did you do?

13b
2 marks

The graph y equals a over x squared passes through the point open parentheses m comma space m to the power of 4 close parentheses, where m is a negative real constant.

Explain whether a greater than 0 or a less than 0.

How did you do?

14
3 marks

Sketch the curve

space y equals 3 x cubed minus 2 x squared minus x

showing the coordinates of any points at which the curve meets the coordinate axes.

How did you do?

15a
3 marks

On the same coordinate axes, sketch the curves y equals 1 over x squared and y equals negative 3 over x squared.

How did you do?

15b
2 marks

Write down the equation(s) of any lines of symmetry and any asymptotes for the two curves in part (a).

How did you do?

1a
2 marks
Graph of a positive cubic curve C which intersects the x axis at the origin and a minimum point touches the x axis at the point x=6. The maximum point with coordinates (2, 8) is marked on the graph.
Figure 1

Figure 1 shows a sketch of a curve C with equation y equals straight f left parenthesis x right parenthesis where straight f left parenthesis x right parenthesis is a cubic expression in x.

The curve

  • passes through the origin

  • has a maximum turning point at left parenthesis 2 comma space 8 right parenthesis

  • has a minimum turning point at left parenthesis 6 comma space 0 right parenthesis

The line with equation y equals k, where k is a constant, intersects C at only one point.

Find the set of values of k, giving your answer in set notation.

How did you do?

1b
3 marks

Find the equation of C. You may leave your answer in factorised form.

How did you do?

2
5 marks

Given that left parenthesis x plus 1 right parenthesis is a factor of x cubed minus 4 x squared plus x plus 6, sketch the curve

space y equals x cubed minus 4 x squared plus x plus 6

showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

3a
4 marks

On the same diagram, sketch the curves with equations y equals x cubed minus 2 x squared minus 8 x and y equals 1 over x.

Show the coordinates of any points where the curves meet the coordinate axes.

How did you do?

3b
1 mark

Hence find the number of solutions to the equation

x cubed minus 2 x squared minus 8 x equals 1 over x

How did you do?

4a
3 marks

By showing that left parenthesis x plus 3 right parenthesis is a factor of the function 

straight f left parenthesis x right parenthesis equals 6 x cubed plus 23 x squared plus 11 x minus 12

fully factorise straight f left parenthesis x right parenthesis.

How did you do?

4b
3 marks

Sketch the curve y equals straight f left parenthesis x right parenthesis, showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

5a
4 marks

Find the coordinates of the points of intersection between the curve with equation 

y equals x cubed minus x squared minus 4 x plus 4

and the line with equation

y equals 2 x plus 4

How did you do?

5b
5 marks

On the same coordinate axes, sketch y equals x cubed minus x squared minus 4 x plus 4 and y equals 2 x plus 4.

Show the coordinates of

  • any points where the curve meets the coordinate axes

  • any points where the line meets the coordinate axes

  • any points of intersection between the curve and the line

How did you do?

1a
2 marks
Graph with two intersecting curves, C1 and C2, on an xy-plane. C1 is ascending in quadrant 1 only from just under half way up the y axis, while C2 is a downward (negative) parabola. C1 and C2 intersect twice
Figure 4

Figure 4 shows a sketch of part of the curve C subscript 1 with equation

y equals 2 x cubed plus 10 space space space space space space space space space space space space space space space space space space x greater than 0

and part of the curve C subscript 2 with equation

y equals 42 x minus 15 x squared minus 7 space space space space space space space space space space space space space space space space space space x greater than 0

Verify that the curves intersect at space x equals 1 half

How did you do?

1b
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5 marks

The curves intersect again at the point P

Using algebra and showing all stages of working, find the exact x coordinate of P

How did you do?

2
5 marks

You are given that left parenthesis x minus 2 right parenthesis and open parentheses x plus 3 close parenthesesare factors of straight f left parenthesis x right parenthesis, where

straight f left parenthesis x right parenthesis equals x to the power of 4 minus 9 x cubed plus 9 x squared plus 85 x minus 150

Sketch the curve y equals straight f left parenthesis x right parenthesis showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

3
4 marks

The curve C has the equation

space y equals 3 x cubed plus 2 x squared minus 3 x plus 10

It is known that

  • the x-coordinate of the minimum point is positive

  • the maximum point is in the second quadrant

By finding a factor of 3 x cubed plus 2 x squared minus 3 x plus 10, sketch C.

Show the coordinates of any points at which the curve meets the coordinate axes.

How did you do?

4a
3 marks

Fully factorise

4 x cubed plus 17 x squared plus 20 x plus 4

How did you do?

4b
3 marks

Sketch the curve

space y equals 4 x cubed plus 17 x squared plus 20 x plus 4

showing the coordinates of the points at which the curve meets the coordinate axes.

How did you do?

5
8 marks

On the same coordinate axes, sketch the curve 

4 y equals x cubed minus 5 x squared minus 12 x plus 36

and the line

x plus y minus 6 equals 0

Show the coordinates of

  • any points where the curve meets the coordinate axes

  • any points where the line meets the coordinate axes

  • any points of intersection between the curve and the line

How did you do?