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Further Graphs & Tangents (Cambridge O Level Maths)

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Further Graphs & Tangents

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Further Graphs & Tangents

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1
Sme Calculator3 marks

cie-igcse-2020-may-jun-p4-tz3-q10b

The diagram shows the graph of another function.
 
By drawing a suitable tangent, find an estimate for the gradient of the function at the point P.

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2a
Sme Calculator2 marks

Write down the equation of your tangent in the form y equals m x plus c.


y equals .................................................

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2b
Sme Calculator3 marks

cie-igcse-2020-march-p4-tz2-q2b

i)
Draw the tangent to the curve at x equals 1.


[1]

ii)
Use your tangent to estimate the gradient of the curve at x equals 1.


[2]

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3
Sme Calculator3 marks

cie-igcse-2019-march-p2-q16

By drawing a suitable tangent, estimate the gradient of the curve at the point P.

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4a
Sme Calculator2 marks

q18-specimen-paper1-2025-4024-01-cambridge-o-level-maths-d

The diagram shows a sketch of the graph of y space equals space 3 space cross times space 2 to the power of x space plus space 5 space.

The graph crosses the y-axis at point A.
Find the coordinates of point A.

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4b
Sme Calculator1 mark

Write down the equation of the asymptote to the graph of  y space equals space 3 space cross times space 2 to the power of x space plus space 5 space.

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5a
Sme Calculator2 marks

The speed–time graph shows the first 10 seconds of the journey of a car.

q13-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

By drawing a tangent, estimate the gradient of the curve at t space equals space 2.

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5b
Sme Calculator1 mark

Write down what the gradient of the curve represents in terms of the car’s motion for the first 10 seconds of its journey.

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6
Sme Calculator2 marks

Sketch the graph of the function y equals 1 over x

q16b-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

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1a
Sme Calculator2 marks

space space space space space y space equals space x to the power of 2 space end exponent plus 1 over x comma space space space x space not equal to space 0

Complete the table.

x 0.2 0.3 0.5 1 1.5 2 2.5
y 5.0 3.4 2.3   2.9   6.7

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1b
Sme Calculator4 marks

On the grid, draw the graph of y space equals space x to the power of 2 space end exponent space plus space 1 over x for 0.2 space less or equal than space x space less or equal than space 2.5.
The graph of y space equals space x to the power of 2 space end exponent space plus space 1 over x for negative 2.5 space less or equal than space x space less or equal than space minus 0.2 has been drawn for you.

cie-igcse-2020-oct-nov-p4-tz2-q7b

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1c
Sme Calculator3 marks

By drawing suitable straight lines on the grid, solve the following equations.

i)blank over blank
x to the power of 2 space end exponent plus space 1 over x space equals space minus 2
 
x space equals ................................................ [1]
ii)blank over blank
x to the power of 2 space end exponent plus space 1 over x space plus space x space minus space 1 equals space 0
 
x space equals ................................................ [2]

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1d
Sme Calculator1 mark

k is an integer and the equation x to the power of 2 space end exponent plus space 1 over x space equals space k has three solutions.

Write down a possible value of k.


k space equals ................................................

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2a
Sme Calculator3 marks

The table shows some values of  y equals x squared over 2 plus 1 over x squared minus 2 over x, x not equal to 0.


x negative 3 negative 2 negative 1 negative 0.5 negative 0.3   0.2 0.3 0.5 1 2 3
y 5.3 3.3   8.1 17.8     4.5 0.1 negative 0.5 1.3  
 
Complete the table.

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2b
Sme Calculator5 marks

On the grid, draw the graph of  y equals x squared over 2 plus 1 over x squared minus 2 over x  for  negative 3 less or equal than x less or equal than negative 0.3  and  0.2 less or equal than x less or equal than 3.

56XoM-7K_cie-igcse-2019-oct-nov-p4-tz2-q5b

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2c
Sme Calculator2 marks

Use your graph to solve  x squared over 2 plus 1 over x squared minus 2 over x less or equal than 0.

..................... less or equal than x less or equal than ....................

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2d
Sme Calculator1 mark

Find the smallest positive integer value of k for which  x squared over 2 plus 1 over x squared minus 2 over x equals k has two solutions for  negative 3 less or equal than x less or equal than negative 0.3  and  0.2 less or equal than x less or equal than 3.

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2e
Sme Calculator3 marks

By drawing a suitable straight line, solve  x squared over 2 plus 1 over x squared minus 2 over x equals 3 x plus 1  for  negative 3 less or equal than x less or equal than negative 0.3  and  0.2 less or equal than x less or equal than 3.

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

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3a
Sme Calculator3 marks

The table shows some values for y equals x cubed plus x squared minus 5 x.

x negative 3 negative 2 negative 1.5 negative 1 0 1 1.5 2 2.5 3
y negative 3 6 6.4   0   negative 1.9 2 9.4  

 

Complete the table.

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3b
Sme Calculator4 marks

On the grid, draw the graph of  y equals x cubed plus x squared minus 5 x  for  negative 3 less or equal than x less or equal than 3.

l8xiBxEy_cie-igcse-2019-oct-nov-p4-tz3-q3b

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3c
Sme Calculator2 marks

Use your graph to solve the equation  x cubed plus x squared minus 5 x equals 0.

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

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3d
Sme Calculator3 marks

By drawing a suitable tangent, find an estimate of the gradient of the curve at x equals 2.

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3e
Sme Calculator1 mark

Write down the largest value of the integer, k, so that the equation  x cubed plus x squared minus 5 x equals k  has three solutions for  negative 3 less or equal than x less or equal than 3.

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

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4a
Sme Calculator3 marks

The table shows some values for space y equals 3 over 10 x cubed minus 2 x for space minus 3 less or equal than x less or equal than 3.

x -3 -2 -1.5 -1 0 1 1.5 2 3
y     2.0 1.7 0   -2.0 -1.6  

Complete the table.

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4b
Sme Calculator4 marks

On the grid, draw the graph of space y equals 3 over 10 x cubed minus 2 x for negative 3 less or equal than x less or equal than 3.

cie-igcse-2019-march-p4-q5

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4c
Sme Calculator4 marks

On the grid, draw a suitable straight line to solve the equation space 3 over 10 x cubed minus 2 x equals 1 half open parentheses 1 minus x close parentheses for negative 3 less or equal than x less or equal than 3.
 

x equals....................... or x equals....................... or x equals......................

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4d
Sme Calculator1 mark

For negative 3 less or equal than x less or equal than 3, the equation space 3 over 10 x cubed minus 2 x equals 1 space has n solutions.
 
Write down the value of n.
 

n equals..................................................

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5a
Sme Calculator3 marks

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

x -0.75 -0.5 -0.25 0 0.5 1 1.5 2 2.5 2.75
y -2.9 -1.4 -0.5   -0.1 -1 -1.9   -0.6  

 

Complete the table.

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5b
Sme Calculator3 marks

On the grid, draw the graph of  y equals x cubed minus 3 x squared plus x  for  negative 0.75 less or equal than x less or equal than 2.75.

6TC-Enhq_cie-igcse-2018-oct-nov-p4-tz1-q3b

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5c
Sme Calculator3 marks

Use your graph to complete the inequalities in x for which y greater than negative 1.
 

.................... less than x less than .................... and  x greater than ....................

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5d
Sme Calculator5 marks

The equation  x cubed minus 3 x squared plus 2 x minus 1 equals 0 can be solved by drawing a straight line on the grid.

i)
Write down the equation of this line.
 

[2]

ii)
On the grid, draw this line and use it to solve the equation  x cubed minus 3 x squared plus 2 x minus 1 equals 0. 
  
 

x = ............................................... [3]

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5e
Sme Calculator3 marks

By drawing a suitable tangent, find an estimate for the gradient of the graph of  y equals x cubed minus 3 x squared plus x  at  x equals negative 0.25.

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6a
Sme Calculator3 marks

Complete the table of values for y equals x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction comma space x not equal to 0.

x negative 3 negative 2 negative 1 negative 0.5 negative 0.3   0.3 0.5 1 2 3
y negative 9.1 negative 2.8 negative 0.8   negative 5.6   negative 5.5 negative 2.0     8.9

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6b
Sme Calculator5 marks

On the grid, draw the graph of y equals x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction for negative 3 less or equal than x less or equal than negative 0.3 and 0.3 less or equal than x less or equal than 3.

cie-igcse-2018-may-jun-2-p4-q6b

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6c
Sme Calculator5 marks
i)
By drawing a suitable tangent, find an estimate of the gradient of the curve at x equals negative 2.

[3]

ii)
Write down the equation of the tangent to the curve at x equals negative 2.
Give your answer in the form y equals m x plus c.

y equals ................................................ [2]

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6d
Sme Calculator4 marks

Use your graph to solve the equations.

i)table row blank row blank end table
x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction equals 0

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

ii)begin mathsize 16px style table row blank row blank end table end style
x cubed over 3 minus fraction numerator 1 over denominator 2 x squared end fraction plus 4 equals 0

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

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7a
Sme Calculator2 marks

The table shows some values for space y space equals space x cubed over 3 – space 2 x space plus space 5 spacecorrect to 1 decimal place.

x –3 –2.5 –2 –1 0 1 2 2.5 3
y   4.8 6.3 6.7 5.0 3.3 3.7 5.2  

Complete the table.

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7b
Sme Calculator4 marks

Draw the graph of y space equals space x cubed over 3 space – space 2 x space plus space 5 space for space – 3 space less-than or slanted equal to space x space less-than or slanted equal to space 3.

q21b-specimen-paper2-2025-4024-02-cambridge-o-level-maths-d

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7c
Sme Calculator4 marks

By drawing a suitable line on the grid, find the roots of the equation x cubed over 3 space – space 1.5 x space plus space 1 space equals space 0.

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8
Sme Calculator6 marks

q7c-june-2021-4024-21-cambridge-o-level-maths-d

The grid shows the graph ofy equals x cubed minus 2 x squared minus 5 x plus 6.

i)
x cubed minus 2 x squared minus 5 x plus 6 equals k has exactly 2 solutions.

Use the graph to find the possible values of k.

[2]

ii)
By drawing a suitable line on the grid, find the solutions of  x cubed minus 2 x squared minus 7 x plus 5 equals 0
x = .................. , x = .................. , x = .................. [4]

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1a
Sme Calculator3 marks

The table shows some values of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 for 0.15 less or equal than x less or equal than 3.5 space.

x 0.15 0.2 0.5 1 1.5 2 2.5 3 3.5
y 3.30   0.88   -0.04   -0.43 -0.58 -0.73

Complete the table. 

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1b
Sme Calculator4 marks

On the grid, draw the graph of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 for 0.15 less or equal than x less or equal than 3.5 space.
The last two points have been plotted for you.

cie-igcse-2019-may-jun-p4-tz2-q5b

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1c
Sme Calculator1 mark

Use your graph to solve the equation fraction numerator 1 over denominator 2 x end fraction minus x over 4 equals 1 half for 0.15 less or equal than x less or equal than 3.5 space. 
 

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

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1d
Sme Calculator4 marks
i)
On the grid, draw the line y equals 2 minus x space. 
 
[2]
ii)
Write down the x co-ordinates of the points where the line y equals 2 minus x crosses the graph
of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 for 0.15 less or equal than x less or equal than 3.5.
 
x = .................... and x = .................... [2]

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1e
Sme Calculator2 marks

Show that the graph of y equals fraction numerator 1 over denominator 2 x end fraction minus x over 4 can be used to find the value of square root of 2 for 0.15 less or equal than x less or equal than 3.5 space.

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2a
Sme Calculator4 marks

The diagram shows the graph of y equals f open parentheses x close parentheses where f open parentheses x close parentheses equals x squared minus 2 over x minus 2 comma space x not equal to 0.

cie-igcse-2019-may-jun-p4-tz3-q5a

On the grid, draw a suitable straight line to solve the equation
x squared minus 2 over x minus 7 equals negative 3 x for negative 3 less or equal than x less or equal than 3.
 

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

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2b
Sme Calculator3 marks

By drawing a suitable tangent, find an estimate of the gradient of the curve at x equals negative 2.

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2c
Sme Calculator7 marks
i)
Complete the table for y equals text g end text left parenthesis x right parenthesis where text g end text left parenthesis x right parenthesis equals 2 to the power of negative x end exponent for negative 3 less or equal than x less or equal than 3.
x -3 -2 -1 0 1 2 3
y     2 1 0.5   0.125

 

[3]

ii)
On the grid, draw the graph of y equals text g end text left parenthesis x right parenthesis.
 
[3]
iii)
Use your graph to find the positive solution to the equation text f end text left parenthesis x right parenthesis equals text g end text left parenthesis x right parenthesis.
 
x = .............................................. [1]

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3a
Sme Calculator7 marks

The graph of y equals 10 minus 8 x squared for  negative 1.5 less or equal than x less or equal than 1.5  is drawn on the grid.

cie-igcse-2018-may-jun-1-7

And the table shows some values for y equals x cubed plus 3 x plus 4.

x negative 1.5 negative 1 negative 0.5 0 0.5 1 1.5
y negative 3.9       5.6 8 11.9

i)
Complete the table.


[3]

ii)
On the grid, draw the graph of y equals x cubed plus 3 x plus 4 for negative 1.5 less or equal than x less or equal than x 1.5.


[4]

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3b
Sme Calculator1 mark

Show that the values of x where the two curves intersect are the solutions to the equation x cubed plus 8 x squared plus 3 x minus 6 equals 0.

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3c
Sme Calculator3 marks

By drawing a suitable straight line, solve the equation x cubed plus 5 x plus 2 equals 0  for  negative 1.5 less or equal than x less or equal than 1.5.

x equals..............................................

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4a
Sme Calculator3 marks

The table shows some values for space y equals 2 x plus 1 over x minus 3 space for  0.125 less or equal than x less or equal than 3.

x 0.125 0.25 0.375 0.5 0.75 1 1.5 2 2.5 3
space y space 5.25 1.5 0.42     0 0.67 1.5   3.33

Complete the table.

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4b
Sme Calculator4 marks

On the grid, draw the graph of space y equals 1 half x plus 1 over x minus 3 space for 0.125 less or equal than x less or equal than 3.

cie-igcse-2018-march-p4-q3b

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4c
Sme Calculator3 marks

Use your graph to solve space 2 x plus 1 over x minus 3 greater or equal than 2.
 

................................................. 
 ................................................. 

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4d
Sme Calculator5 marks

The equation space 1 over x equals 7 minus 3 x space can be solved using your graph in part (b) and a straight line.

i)

Write down the equation of this straight line.
 

[2]

ii)table row blank row blank end table

Draw this straight line and solve the equation space 1 over x equals 7 minus 3 x.
 

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

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5
Sme Calculator8 marks

The table shows some values of x and the corresponding values of y for space y space equals space 2 x cubed space – space 3 x squared space plus space 5.

x negative 1.5 negative 1 negative 0.5 0 0.5 1 1.5 2
y   0 4 5 4.5 4 5 9

i)
Complete the table. 

[1]

ii)
Using a scale of 4 cm to represent 1 unit, draw a horizontal x-axis for – 1.5 space less or equal than space x space less or equal than space 2.
Using a scale of 2 cm to represent 5 units, draw a vertical y-axis for – 10 space less or equal than space y space less or equal than space 10.

Draw the graph of y space equals space 2 x cubed space – space 3 x squared space plus space 5 for – 1.5 space less or equal than space x space less or equal than space 2.
tT_F1C0y_q6aii-june-2018-4024-2-cambridge-o-level-maths-d

[3]

iii)
Use your graph to estimate the gradient of the curve when x space equals space 1.5.

[2]

iv)
By drawing a suitable line on your graph, find the solution of the equation 2 x cubed space – space 3 x squared space plus space 4 space equals space 0.

x = ...................[2]

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6
Sme Calculator1 mark

q6b-june-2018-4024-2-cambridge-o-level-maths-d

The graph shows a sketch of the curve y space equals space p over x
Two points on the curve are left parenthesis 3 comma space 0.4 right parenthesis and left parenthesis q comma space 2.4 right parenthesis

i)
Find p and q.

p = ...................
q = ...............[2]

ii)
Calculate the gradient of the straight line joining the points left parenthesis 3 comma space 0.4 right parenthesis and left parenthesis q comma space 2.4 right parenthesis.

[2]

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