ASMaths: PureOCRExam Questions2. Algebra & Functions2.6 Graphs of Functions

Graphs of Functions (OCR AS Maths: Pure)

Exam Questions

3 hours40 questions
Easy
Medium
Hard
Very Hard
1
Sme Calculator3 marks

Sketch the graph of  space y equals 6 x minus 12, labelling any points where the graph intersects the coordinate axes.

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2
Sme Calculator3 marks

Sketch the graph of space y equals x squared minus 1, labelling any points where the graph intersects the coordinate axes.

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3
Sme Calculator3 marks

Sketch the graph of space y equals 1 over x, labelling any points where the graph intersects the coordinate axes and stating the equations of any asymptotes.

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4
Sme Calculator3 marks
(i)

On the axes below, sketch the graphs of both space y equals x space and space y equals space minus x plus 2.

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

 
(ii)

Using your graph, or otherwise, find the solution to the simultaneous equations

space y equals x spaceand space y equals space minus x plus 2.

 

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5a
Sme Calculator2 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 

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5b
Sme Calculator3 marks

Sketch the graph of space y equals x squared plus 3 x minus 4, labelling any points where the graph intersects the coordinate axes.

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6a
Sme Calculator2 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 comma space b space and space c are integers to be found.

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6b
Sme Calculator3 marks

Hence sketch the graph of space y equals 2 x cubed plus 2 x squared minus 12 x labelling any points where the graph intersects the coordinate axes.

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7
Sme Calculator3 marks

space y is proportional to x.

When x equals 2 comma space y equals 10.

(i)

Find the constant of proportionality

(ii)

Sketch the graph of space y against x.

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8
Sme Calculator3 marks

By sketching the graphs of y equals x cubed and space y equals 1 over x on the same diagram show that there are two real solutions to the equation x cubed equals 1 over x.

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9a
Sme Calculator2 marks

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.

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9b
Sme Calculator2 marks

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

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9c
Sme Calculator3 marks

Sketch the graph of space y equals straight f left parenthesis x right parenthesis labelling any points where the graph intersects the coordinate axes.

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10
Sme Calculator3 marks

The diagram below shows the graph of a circle with equation left parenthesis x minus 3 right parenthesis squared plus left parenthesis y plus 2 right parenthesis squared equals 9.

Add straight lines passing through the point to the diagram to show how the circle can have either no, one or two intercepts. All lines must pass through the point left parenthesis negative 6 comma negative 4 right parenthesis.

2-7-edexcel-alevel-maths-pure-q10easy

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1
Sme Calculator3 marks

Sketch the graph of y equals left parenthesis x plus 3 right parenthesis cubed labelling any points where the graph intersects the coordinate axes.

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2
Sme Calculator2 marks

Sketch the graph of space y equals 1 over x squared and write down the equations of any asymptotes.

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3a
Sme Calculator2 marks

On the axes below sketch the graphs of both space y equals 3 x minus 2 space and space y equals x plus 4.

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

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3b
Sme Calculator1 mark

Using your graph, or otherwise, find the solution to the simultaneous equations

3 x minus y equals 2 space and
x minus y equals negative 4.

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4
Sme Calculator3 marks

space y is inversely proportional to x.  When x equals 3 comma space y equals 12.  Find the constant of proportionality and sketch the graph of space y against x.

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5
Sme Calculator3 marks

Sketch the graph of y equals 2 x squared left parenthesis x plus 3 right parenthesis labelling any points where the graph intersects the coordinate axes.

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6a
Sme Calculator3 marks

On the same diagram, sketch the graphs of space y equals x left parenthesis x plus 2 right parenthesis left parenthesis x minus 1 right parenthesis space and space y equals 1 over x.

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6b
Sme Calculator1 mark

Use your graph to determine the number of solutions to the equation

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

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7a
Sme Calculator2 marks

A machine computes a calculation in time, t seconds, that is proportional to the number of processes, space p, involved.  For a calculation involving 10 processes the machine takes 0.01 seconds.

Show that the constant of proportionality is 0.001.

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7b
Sme Calculator2 marks

Hence write down an equation linking the number of processes open parentheses p close parentheses to the time taken open parentheses t close parentheses.

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7c
Sme Calculator2 marks

Find the time it takes for the machine to compute a calculation involving 200 processes.

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7d
Sme Calculator2 marks

How many processes are involved for a calculation taking 2.3 seconds?

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8a
Sme Calculator2 marks

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

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

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

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8b
Sme Calculator4 marks

State which graph the following points must lie on and find the value of a in each case.

(i)
left parenthesis negative 4 comma negative 5 right parenthesis 
(ii)
left parenthesis negative 0.02 comma space 250 right parenthesis

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9a
Sme Calculator1 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.

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9b
Sme Calculator2 marks

Fully factorise 2 x cubed minus 3 x squared minus 11 x plus 6.

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9c
Sme Calculator4 marks

Sketch the graph of space y equals 2 x cubed minus 3 x squared minus 11 x plus 6.
Label any points where the graph crosses the coordinate axes.

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10a
Sme Calculator3 marks

Solve the equation x cubed minus x squared minus 2 x plus 4 equals 4 x plus 4.

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10b
Sme Calculator2 marks

Write down the x-coordinates of the points of intersection between the graphs of space y equals x cubed minus x squared minus 2 x plus 4 and space y equals 4 x plus 4.

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10c
Sme Calculator2 marks

Find the space y-coordinates of the points of intersection between the graphs of space y equals x cubed minus x squared minus 2 x plus 4 and space y equals 4 x plus 4.

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10d
Sme Calculator4 marks

On the same diagram, sketch the graphs of space y equals x cubed minus x squared minus 2 x plus 4 and space y equals 4 x plus 4.

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1
Sme Calculator4 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 graph of space y equals x cubed minus 4 x squared plus x plus 6. Label any points where the graph intersects the coordinate axes.

(There is no need to label any stationary points.)

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2
Sme Calculator2 marks

Sketch the graph of space y equals fraction numerator negative 2 over denominator x squared end fraction and write down the equations of any asymptotes.

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3a
Sme Calculator3 marks

On the axes below sketch the graphs of both space y equals x squared plus 2 x minus 3 and space y equals x minus 1.

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

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3b
Sme Calculator2 marks

Using your graph, or otherwise, find the solutions to the simultaneous equations

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

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4
Sme Calculator4 marks

space y is inversely proportional to x.  When x equals 2 comma space y equals 10.  Find the constant of proportionality and sketch the graph of space y against x.

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5
Sme Calculator3 marks

Sketch the graph of space y equals 3 x cubed minus 2 x squared minus x labelling any points where the graph intersects the coordinate axes.

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6a
Sme Calculator3 marks

On the same diagram, sketch the graphs of space y equals x cubed minus 2 x squared minus 8 x and space y equals 1 over x.

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6b
Sme Calculator1 mark

Use your graph to determine the number of solutions to the equation

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

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7a
Sme Calculator3 marks

A machine computes a calculation in time, t seconds, that is proportional to the square of the number of processes, space p, involved.  For a calculation involving 8 processes the machine takes 0.032 seconds.

Find an equation linking the number of processes, (space p) to the time taken, (t) seconds.

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7b
Sme Calculator2 marks

How many processes are involved for a calculation taking 0.2 seconds?

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7c
Sme Calculator2 marks

Find the time it takes for the machine to compute a calculation involving 30 processes.

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8a
Sme Calculator2 marks

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

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

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

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8b
Sme Calculator2 marks

Given that m is a negative real number, state with a reason, which graph passes through the point open parentheses m comma space m to the power of 4 close parentheses.

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9a
Sme Calculator3 marks

Show 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 and hence, or otherwise, fully factorise straight f left parenthesis x right parenthesis.

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9b
Sme Calculator4 marks

Sketch the graph of space y equals straight f left parenthesis x right parenthesis.
Label any points where the graph crosses the coordinate axes.

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10a
Sme Calculator4 marks

Find the coordinates of the points of intersection between the curve with equation space y equals x cubed minus x squared minus 4 x plus 4 and the line with equation space y equals 2 x plus 4.

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10b
Sme Calculator5 marks

On the same diagram, sketch the graphs of space y equals x cubed minus x squared minus 4 x plus 4 and space y equals 2 x plus 4.
Label the coordinates of any points of intersection between the two graphs.
Also label any points where the graphs intersect the coordinate axes.

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1
Sme Calculator5 marks

left parenthesis x minus 2 right parenthesis space space and space left parenthesis x plus 3 right parenthesis are 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 graph of space y equals straight f left parenthesis x right parenthesis labelling any points where the graph intersects the coordinate axes.  (There is no need to label any stationary points).

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2a
Sme Calculator3 marks

On the same diagram, sketch the graphs of space y equals 1 over x squared and space y equals fraction numerator negative 3 over denominator x squared end fraction.

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2b
Sme Calculator2 marks

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

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3a
Sme Calculator3 marks

On the axes below sketch the graphs of both space y equals left parenthesis x minus 1 right parenthesis squared and space y equals 2 minus x squared minus x.

2-7-edexcel-alevel-maths-pure-q3vhard

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3b
Sme Calculator2 marks

Using your graph, or otherwise, find the solutions to the equation

x squared minus 2 x plus 1 equals 2 minus x squared minus x.

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4
Sme Calculator4 marks

space y is inversely proportional to the square of x.  When x equals 4 comma space y equals 8.  Find the constant of proportionality and sketch the graph of space y against x.

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5
Sme Calculator4 marks

Sketch the graph of space y equals 3 x cubed plus 2 x squared minus 3 x plus 10 labelling any points where the graph intersects the coordinate axes.

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6a
Sme Calculator4 marks

On the same diagram, sketch the graphs of space y equals x cubed minus 3 x squared minus 6 x plus 8 and space y equals 3 over x squared.

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6b
Sme Calculator1 mark

Write down the number of solutions to the equation

x to the power of 5 minus 3 x to the power of 4 minus 6 x cubed plus 8 x squared equals 3.

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7a
Sme Calculator4 marks

A machine computes a calculation in time, t seconds, that is proportional to the cube root of the number of processes, space p, involved.  For a calculation involving 8 processes the computer takes 6.4 space cross times 10 to the power of negative 4 end exponent seconds.

How many processes are involved for a calculation taking 1.28 cross times 10 to the power of negative 3 end exponent seconds?

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7b
Sme Calculator2 marks

Find the time it takes for the machine to compute a calculation involving 250 processes.

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8a
Sme Calculator3 marks

On separate diagrams, sketch the graphs of space y equals a over x squared, where a greater than 0 and space y equals a over x squared, where a less than 0.

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8b
Sme Calculator3 marks

One of the graphs passes through the point with coordinates open parentheses m comma space m to the power of 6 close parentheses.

Write a in terms of m and, justifying your answer, state which graph this point must lie on.

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9a
Sme Calculator3 marks

Fully factorise 4 x cubed plus 17 x squared plus 20 x plus 4.

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9b
Sme Calculator2 marks

Sketch the graph of space y equals 4 x cubed plus 17 x squared plus 20 x plus 4.
Label any points where the graph crosses the coordinate axes.

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10
Sme Calculator9 marks

On the same diagram, sketch the graphs of 4 y equals x cubed minus 5 x squared minus 12 x plus 36 and x plus y minus 6 equals 0.

Label the coordinates of any points of intersection between the two graphs.
Also label any points where the graphs intersect the coordinate axes.

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