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Further Graphs & Tangents (Cambridge O Level Maths)

Exam Questions

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Further Graphs & Tangents

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Further Graphs & Tangents

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1
Sme Calculator4 marks

On the diagram, sketch the graph of y space equals space open parentheses x space plus space 1 close parentheses open parentheses x space minus space 3 close parentheses squared.
Label the values where the graph meets the x-axis and the y-axis.

cie-igcse-2020-oct-nov-p4-tz3-q7b

 

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2
Sme Calculator2 marks

Sketch the graph of the function.

y equals 1 over x

cie-igcse-2020-march-p2-tz2-q10a-b

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3
Sme Calculator3 marks

The diagrams A, B, C, D, E and F are six graphs of different functions.

cie-igcse-q21-graphs-diagrams

Complete the table to identify the correct graph for each function.
One has been done for you. 

Function y space equals space x space plus space 1 y space equals space 1 space minus space x over 3 y space equals space 2 x squared y space equals space minus 4 over x
Diagram E      

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1a
Sme Calculator3 marks

q26-paper2-spec2025-cie-igcse-further-maths

straight f left parenthesis x right parenthesis space equals space x left parenthesis x space plus space 2 right parenthesis left parenthesis x space – space 3 right parenthesis space

On the diagram, sketch the graph of y space equals space straight f left parenthesis x right parenthesis space space space space space space for space – 3 space less-than or slanted equal to space x space less-than or slanted equal to space 4 .
Show the values of the intersections with the axes. 

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1b
Sme Calculator3 marks

Expand and simplify.

x left parenthesis x space plus space 2 right parenthesis left parenthesis x space – space 3 right parenthesis

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2a
Sme Calculator4 marks

The diagram shows the graph of y space equals space straight f open parentheses x close parentheses for  negative 3 space less or equal than space x space less or equal than space 3.

cie-igcse-2020-oct-nov-p4-tz1-q5a

i)
Solve space straight f left parenthesis space x right parenthesis space equals space 14. space

space x space equals space................................................ [1]
ii)
By drawing a suitable tangent, find an estimate of the gradient of the graph at the point left parenthesis negative 2 comma space 4 right parenthesis.
 
[3]

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2b
Sme Calculator3 marks

By drawing a suitable straight line on the grid, solvespace space straight f left parenthesis space x right parenthesis space equals space 2 x space minus space 2 space for negative 3 space less or equal than space x space less or equal than space 3.

space x space equals space................................................

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3a
Sme Calculator3 marks

The table shows some values for y equals 2 x cubed minus 4 x squared plus 3.

x -1 -0.5 0 0.5 1 1.5 2
y -3 1.75       0.75 3

Complete the table.

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3b
Sme Calculator4 marks

Draw the graph of y equals 2 x cubed minus 4 x squared plus 3 for negative 1 less or equal than x less or equal than 2.


cie-igcse-2020-march-p4-tz2-q2a

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3c
Sme Calculator4 marks
i)
Use your graph to solve the equation 2 x cubed minus 4 x squared plus 3 equals 1.5.


x equals ........................ or x equals ........................ or x equals ........................ [3]

ii)
The equation 2 x cubed minus 4 x squared plus 3 equals k has only one solution for negative 1 less or equal than x less or equal than 2.

Write down a possible integer value of k.


[1]

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4a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis equals x squared over 4 minus 4 over x, x not equal to 0
 
Complete the table for straight f left parenthesis x right parenthesis.

x 0.5 1 2 3 4 5 6
straight f left parenthesis x right parenthesis -7.9 -3.8   0.9   5.5 8.3

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4b
Sme Calculator3 marks

The graph of y equals straight f left parenthesis x right parenthesis for negative 6 less or equal than x less or equal than negative 0.5 is drawn on the grid.

cie-igcse-2018-oct-nov-p4-tz3-q4b

On the same grid, draw the graph of y equals straight f left parenthesis x right parenthesis for 0.5 less or equal than x less or equal than 6.

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4c
Sme Calculator3 marks

By drawing a suitable tangent, estimate the gradient of the graph of y equals straight f left parenthesis x right parenthesis at the
point (–4, 5).

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4d
Sme Calculator1 mark

straight g left parenthesis x right parenthesis equals 9 over x,  x not equal to 0
 
Complete the table for straight g left parenthesis x right parenthesis.
 

x -4 -3 -2 -1   1 2 3 4
straight g left parenthesis x right parenthesis -2.3   -4.5 -9   9 4.5   2.3

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4e
Sme Calculator4 marks

On the same grid, draw the graph of  y equals straight g left parenthesis x right parenthesis for  negative 4 less or equal than x less or equal than negative 1  and  1 less or equal than x less or equal than 4.

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4f
Sme Calculator2 marks
i)
Use your graphs to find the value of x when  straight f left parenthesis x right parenthesis equals straight g left parenthesis x right parenthesis.
 
 

x = ................................................. [1]

ii)
Write down an inequality to show the positive values of x for which  straight f left parenthesis x right parenthesis greater than straight g left parenthesis x right parenthesis.
 
 

[1]

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5a
Sme Calculator4 marks
i)
y equals 2 to the power of x

Complete the table

x 0 1 2 3 4
y   2 4 8  

[2]
ii)
y equals 14 minus x squared

Complete the table

x 0 1 2 3 4
y   13 10 5  

[2]

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5b
Sme Calculator6 marks

On the grid, draw the graphs of  y equals 2 to the power of x and  y equals 14 minus x squared  for  0 less or equal than x less or equal than 4.
cie-igcse-2018-may-jun-3-p4-q2b

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5c
Sme Calculator2 marks

Use your graphs to solve the equations.

i)
2 to the power of x equals 12

x equals............................................... [1]

ii)
2 to the power of x equals 14 minus x squared

x equals............................................... [1]

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5d
Sme Calculator3 marks
i)
On the grid, draw the line from the point left parenthesis 4 comma space 2 right parenthesis that has a gradient of negative 4.

[1]

ii)

Complete the statement.

This straight line is a .................................. to the graph of y equals 14 minus x squared
at the point ( ....... , ....... ).

[2]

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1a
Sme Calculator2 marks

straight f left parenthesis x right parenthesis space equals space 20 over x space plus space x comma space space space space x space not equal to space 0

Complete the table.

x -10 -8 -5 -2 -1.6   1.6 2 5 8 10
straight f left parenthesis x right parenthesis -12 -10.5 -9 -12 -14.1   14.1 12     12

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1b
Sme Calculator5 marks

On the grid, draw the graph of  y space equals space straight f left parenthesis x right parenthesis for space – 10 space less-than or slanted equal to space x space less-than or slanted equal to space – 1.6 space and  1.6 space less-than or slanted equal to space x space less-than or slanted equal to space 10.

cie-igcse-spec-p4-q3b-graph

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1c
Sme Calculator2 marks

Using your graph, solve the equation straight f left parenthesis x right parenthesis space equals space 11.

x = ..................   or   x = .................

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1d
Sme Calculator2 marks

k is a prime number and space straight f left parenthesis x right parenthesis space equals space k spacehas no solutions.

Find the possible values of k.

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1e
Sme Calculator1 mark

The gradient of the graph of y space equals space straight f left parenthesis x right parenthesis at the pointspace left parenthesis 2 comma space 12 right parenthesis spaceis negative 4.

Write down the coordinates of the other point on the graph of y space equals space straight f left parenthesis x right parenthesis where the gradient is negative 4.

 (....................... , .......................)

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1f
Sme Calculator6 marks
i)
The equation straight f left parenthesis x right parenthesis space equals space x squared can be written as space x cubed space plus space p x squared space plus space q space equals space 0.

Show that space p space equals space minus 1 spaceand q space equals space minus 20.

[2]
ii)
On the grid, draw the graph of  y space equals space x squared space for  – 4 space less-than or slanted equal to space x space less-than or slanted equal to space 4. 
 
[2]
iii)
Using your graphs, solve the equation space x cubed space space – space x squared space space – space 20 space equals space 0.
 
x = .............................................. [1]
iv)
qJ9LLAKn_cie-igcse-spec-p4-q3fiv-graph
The diagram shows a sketch of the graph of y space equals space x cubed space space – space x squared space space – space 20.
P is the point left parenthesis n comma space 0 right parenthesis.

Write down the value of n.

n space equals .............................................. [1]

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2a
Sme Calculator3 marks

The table shows some values for y equals x cubed plus 3 x squared plus 2.

x -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5
y -4.1   5.1 6 5.4 4 2.6   2.9   12.1

Complete the table.

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2b
Sme Calculator4 marks

On the grid, draw the graph of y equals x cubed plus 3 x squared plus 2 for negative 3.5 less or equal than x less or equal than 1.5 space.

cie-igcse-2019-may-jun-1-q2-2

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2c
Sme Calculator1 mark

Use your graph to solve the equation x cubed plus 3 x squared plus 2 equals 0 for negative 3.5 less or equal than x less or equal than 1.5 space.
 

x = ....................................................

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2d
Sme Calculator2 marks

By drawing a suitable straight line, solve the equation x cubed plus 3 x squared plus 2 x plus 2 equals 0 for negative 3.5 less or equal than x less or equal than 1.5 space.
 

x = ....................................................

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2e
Sme Calculator1 mark

For negative 3.5 less or equal than x less or equal than 1.5 space comma the equation x cubed plus 3 x squared plus 2 equals k has three solutions and k is an integer.
Write down a possible value of k.
 

k = ....................................................

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3a
Sme Calculator2 marks

The table shows some values of  y equals x cubed minus 3 x minus 1.

x -3 -2.5 -2 -1.5 -1 0 1 1.5 2 2.5 3
y -19 -9.1   0.1 1 -1 -3 -2.1 1 7.1  

 
Complete the table of values.

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3b
Sme Calculator4 marks

Draw the graph of  y equals x cubed minus 3 x minus 1  for  negative 3 less or equal than x less or equal than 3.

cie-igcse-2018-oct-nov-p4-tz2-q5b

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3c
Sme Calculator5 marks

A straight line through (0, –17) is a tangent to the graph of  y equals x cubed minus 3 x minus 1.

i)
On the grid, draw this tangent. 
  
 

[1]

ii)
Find the co-ordinates of the point where the tangent meets your graph. 
 
 

(................ , ................) [1]

iii)
Find the equation of the tangent.
Give your answer in the form  y equals m x plus c.
 
 

y = ................................................ [3]

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3d
Sme Calculator4 marks

By drawing a suitable straight line on the grid, solve the equation  x cubed minus 6 x minus 3 equals 0.
 

x = ................... or x = ................... or x = ...................

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4
Sme Calculator4 marks

q25-june-2021-4024-11-cambridge-o-level-maths-d

The diagram shows a circle centre C space left parenthesis 0 comma space 1 right parenthesis.
space P space left parenthesis 3 comma space 5 right parenthesis spaceis a point on the circumference of the circle.

Find the equation of the tangent at P.

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