OCR GCSE Maths
Topic questions Pure mathematics Graphs and curve sketching
26 questions · 161 marks · about 2.5 hours
Also involves Symmetry
Here is a table of values for y = x2 - x - 8.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 12 | 4 | -2 | -6 | -8 | -8 | -6 | -2 | 4 |
Draw the graph of y = x2 - x - 8 for -4 ⩽ x ⩽ 4.
Write down the equation of the line of symmetry of the graph.
Use the graph to solve the equation x2 - x - 8 = 0. Give your answers to 1 decimal place.
Also involves Percentages
The expected value of a painting, £ P, is given by the formula
P = 2500 × 1.2n
where n is the number of years after it was bought and 0 ⩽ n ⩽ 4.
Write down the value of the painting when it was bought.
Write down the annual percentage increase in the expected value of the painting.
The table shows the expected value of the painting n years after it was bought.
| Years after the painting is bought (n) | 1 | 2 | 3 | 4 |
| Expected value of the painting (£) | 3000 | 3600 | 4320 | 5184 |
On the page opposite, draw a suitable graph to show the expected value of the painting n years after it was bought, where 0 ⩽ n ⩽ 4.
Years after the painting was bought (n)
An art collector correctly works out 2500 × 1.210 as 15479.
They say,
The expected value of the painting 10 years after it was bought is £15479.
What assumption has the art collector made.
Here is a table of values for y = 2x2 - 3x.
| x | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 14 | 5 | 0 | -1 | 2 | 9 |
Draw the graph of y = 2x2 - 3x for values of x from 2- to 3.
Use your graph to find the x-coordinates of the points where the graph of y = 2x2 - 3x crosses the line y = 6.
(b) x =........ and x =........
Also involves Substitution into formulae
Complete this table for y = x3 - 3x2.
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | -20 | 0 | -2 | -4 | 16 |
Draw the graph of y = x3 - 3x2 for values of x from 2- to 4.
Use the graph to solve the equation x3 - 3x2 = 5. Give your answer to 1 decimal place.
Here is a table of values for y = 
+ 2.
| x | -6 | -3 | -2 | -1 | 1 | 2 | 3 | 6 |
| y | 1 | 0 | -1 | -4 | 8 | 5 | 4 | 3 |
Draw the graph of y = 
+ 2 for -6 ⩽ x ⩽ 6, x ≠ 0.
Also involves Cubic, reciprocal and exponential graphs
Complete the table for y = x3 + 4x2.
| x | -5 | -4.5 | -4 | -3 | -2 | -1 | 0 | 1 |
| y | -25 | -10.1 | 0 | 9 | 8 | 0 | 5 |
Draw the graph of y = x3 + 4x2 for -5 ⩽ x ⩽ 1.
The equation x3 + 4x2 = k, where k is an integer, has exactly one solution for -5 ⩽ x ⩽ 1.
Find the greatest possible value of k.
Here is a table of values for y = 
- 2x.
| x | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 |
| y | 6.5 | 4 | 1 | -4 | 4 | -1 | -4 | -6.5 |
Draw the graph of y = 
- 2x for 4 ⩽ x ⩽ 4, x ≠ 0.
Use your graph to find the positive solution of 
- 2x = 0. Give your answer to 1 decimal place.
Here is a table of values for y = - 2x.
| x | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 |
| y | 6.5 | 4 | 1 | -4 | 4 | -1 | -4 | -6.5 |
Draw the graph of y = - 2x for 4 ⩽ x ⩽ 4, x ≠ 0.
Use your graph to find the positive solution of 
- 2x = 0. Give your answer to 1 decimal place.
Complete this table for y = x2 - 5.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 4 | -1 | -4 | -4 | -1 | 4 |
Draw the graph of y = x2 - 5 for the values of x from -4 to 3.
Use the graph to solve the equation x2 - 5 = 0. Give your answers to 1 decimal place.
Also involves Rounding
1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.
The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.
Complete the table. Round your answers to the nearest integer.
| Years after release | 0 | 1 | 2 | 3 | 4 |
| Expected number of fish | 1600 | 1680 | 1764 |
Use the table to draw a suitable graph to show the expected number of fish in the lake.
A maximum of 2000 fish can live in the lake.
What effect would you expect this to have on the shape of your graph after 4 years?
TURN OVER FOR QUESTIONS 25 AND 26
Here is a table of values for y = x2 + 2x - 2.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 |
| y | 6 | 1 | -2 | -3 | -2 | 1 | 6 |
Draw the graph of y = x2 + 2x - 2 for -4 ⩽ x ⩽ 2.
Write down the equation of the line of symmetry of the graph.
Use the graph to solve the equation x2 + 2x - 2 = 0. Give your answers to 1 decimal place.
(c) x =........ or x =........
Also involves Rounding
1600 fish are released into a new lake which has no fish. The number of fish is expected to increase by 5% each year.
The table shows the expected number of fish in the lake at the end of 1 year and at the end of 2 years.
Complete the table. Round your answers to the nearest integer.
| Years after release | 0 | 1 | 2 | 3 | 4 |
| Expected number of fish | 1600 | 1680 | 1764 |
Use the table to draw a suitable graph to show the expected number of fish in the lake.
A maximum of 2000 fish can live in the lake.
What effect would you expect this to have on the shape of your graph after 4 years?
Complete this table for y = x2 - 3.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 1 | -2 | -2 | 1 | 6 |
Draw the graph of y = x2 - 3 for values of x from -3 to 3.
Use your graph to solve x2 - 3 = 2.
Complete the table for y = 
| x | -6 | -3 | -2 | -1 | 1 | 2 | 3 | 6 |
| y | -1 | -2 | -6 | 6 | 3 | 2 | 1 |
Draw the graph of y = 
for -6 ⩽ x ⩽ 6, x ≠ 0.
Complete this table for y = x2 - x.
| x | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 6 | 0 | 2 | 6 |
The graph of y = x + 1 is shown on the grid.
On the same grid, use part (a) to draw the graph of y = x2 - x for values of x from -2 to 3.
Write down the x-coordinates of the points where y = x2 - x and y = x + 1 cross.
(c) x =........ and x =........
Complete the table for y = x2 - 4x + 1.
| x | -1 | 0 | 1 | 2 | 3 | 4 | 5 |
| y | 1 | -2 | 6 |
Draw the graph of y = x2 - 4x + 1 for -1 ⩽ x ⩽ 5.
On the same grid, draw the graph of y = 2 x - 6 for - 1 ⩽ x ⩽ 5.
Use your graphs to solve the equation x2 - 4x + 1 = 2x - 6.
Give your answers to 1 decimal place.
Complete this table for y = 4 x - 2.
| x | 0 | 1 | 2 | 3 | 4 |
| y | -2 | 2 | 6 | 14 |
On the grid below, draw the graph of y = 4 x - 2 for values of x from 0 to 4.
The diagram below shows part of another straight line.
Find the equation of this straight line.
Also involves Rearranging formulae · Gradient and intercept
The graph of y = x3 - x2 - 2 is drawn on the grid.
Use the graph to solve x3 - x2 - 2 = 0. Give your answer correct to 1 decimal place.
The equation x3 - x2 + 5x - 6 = 0 can be solved by finding the intersection of the graph of y = x3 - x2 - 2 and the line y = ax + b.
Find the value of a and the value of b.
Hence, use the graph to solve the equation x3 - x2 + 5x - 6 = 0. Give your answer correct to 1 decimal place.
The number of gannets on an island is assumed to follow this exponential growth model.
N is the number of gannets, in thousands. x is the number of years after 1st January 2010.
Complete the table for N = 0.45 × 1.07x.
| x | 0 | 5 | 10 | 15 | 20 |
| N | 0.45 | 0.63 | 1.24 |
Draw the graph of N = 0.45 × 1.07x.
Number of years, x, after 1st January 2010
Use the graph to find the year when the gannet population is predicted to reach 1000.
For each graph below, select its possible equation from this list.
Also involves Cubic, reciprocal and exponential graphs
Complete the table for y = x3 - 3x.
| x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | -18 | -2 | 0 | -2 | 2 | 18 |
Draw the graph of y = x3 - 3x for -3 ⩽ x ⩽ 3.
Use your graph to solve x3 - 3x = 10.
Complete this table for y = x2 - 5.
| x | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
| y | -4 | -5 | -4 | 11 |
On the grid below, draw the graph of y = x2 - 5 for the values of x from -2 to 4.
On the same grid, draw the line y = - 2.
Write down the x coordinates of the points where y = x2 - 5 and y = -2 cross.
(d) x =........ and x =........
A company tests a new battery for an electric car. The distance the car travels, d km, and the charge left in the battery, C %, are measured.
Some measurements are shown in the table.
| Distance travelled, d km. | 0 | 50 | 100 | 150 |
| Charge left in the battery, C %. | 100 | 75 | 50 | 25 |
Plot these values on the grid and use them to draw a straight line.
Use your line to estimate the greatest distance the car will travel.
What assumption is made when estimating the greatest distance?
For your line in part (a), find
the gradient,
the C-axis intercept.
Use your answers to part (c) to write down the equation of your graph.
Give your equation in the form C = ad + b.
Use your equation to find the value of C when d = 210.
Comment on your answer.
Also involves Straight line graphs
Complete this table for y = x2 + x - 4.
| x | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
| y | 2 | -4 | -4 | 2 |
Draw the graph of y = x2 + x - 4 for -4 ⩽ x ⩽ 3.
Use your graph to solve x2 + x - 4 = 0.
On the same grid, draw the graph of y = - 2 x - 1 for - 4 ⩽ x ⩽ 3. You may use the table if you wish.
Use your graphs to solve the equation x2 + x - 4 = -2x - 1.
Complete the table for y = x2 - 2x.
| x | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 3 | 0 | -1 | 0 | 3 |
Draw the graph of y = x2 - 2x for -1 ⩽ x ⩽ 4.
Use your graph to solve x2 - 2x = 2.
Complete the table for y = x2 - 2x.
| x | -1 | 0 | 1 | 2 | 3 | 4 |
| y | 3 | 0 | -1 | 0 | 3 |
Draw the graph of y = x2 - 2x for -1 ⩽ x ⩽ 4.
Use your graph to solve x2 - 2x = 2.
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