37 Interpreting graphs
GCSE Mathematics for OCR (Foundation)
In this chapter you will learn how to …
or more resources relating
F
to this chapter, visit GCSE
Mathematics Online.
• construct and interpret graphs in real-world contexts.
• interpret the gradient of a straight-line graph as a rate of change.
Using mathematics: real-life applications
All sorts of information can be obtained from graphs in real-life contexts.
The shape of a graph, its gradient and the area underneath it can tell us
about speed, time, acceleration, prices, earnings, break-even points or
the values of one currency against another, among other things.
“My car needs to perform at its optimum limits.
We generate and analyse diagnostic graphs to
calculate the slight changes that would increase
power, acceleration and top speed.”
(F1 racing driver)
Before you start …
Ch 34
You will need to be able to
distinguish between direct
and inverse proportion.
1 Which of these graphs shows an inverse proportion?
How do you know this?
A
A
6
6
5
5
4
4
3
3
2
2
1
1
0
00
0
Ch 18
You’ll need to be able to
calculate the gradient of a
straight line.
1
1
2
2
3
3
4
4
5
5
6
6
6
6
5
5
4
4
3
3
2
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1
1
0
00
0
B
B
1
1
2
2
2 Calculate the gradient of AB.
3
3
C
C
4
4
5
5
6
6
5
5
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3
2
2
1
1
0
00
0
6
6
y
5
1
1
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B
4
3
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1
0
578
A
0
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4
5 x
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37 Interpreting graphs
Assess your starting point using the Launchpad
Step 1
1 Describe what is happening in each of the distance–time graphs
below. Suggest a possible real-life situation that would result in
each graph.
a
b
c
d
d
Go to
Section 1:
Graphs of real-world
contexts
Distance
(d)
t
Time (t)
d
e
d
t
f
d
t
d
t
t
✓
Step 2
2 Consider the following graphs showing the journey of a car.
Which four match the situations described below?
Give reasons.
a
b
Distance
c
Distance
Time
d
Distance
Time
e
Speed
Time
f
Speed
Time
Go to
Section 2:
Gradients
Speed
Time
Time
A The car is travelling at a constant speed.
B The car is accelerating at a constant rate.
C The car’s acceleration is increasing.
D The car is stationary.
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
✓
Go to
Chapter review
579
GCSE Mathematics for OCR (Foundation)
Section 1: Graphs of real-world contexts
Graphs are useful for showing the relationships between quantities.
For example, a group of people buy tickets to attend a play at the costs shown
in the graph below. The tickets include transport costs and seats in the hall.
This graph shows lots of information.
The horizontal axis (or x-axis) shows the
number of people attending. The vertical
axis (or y-axis) shows the total cost.
110
100
The cost depends on the number of
people attending. However, there is
a cost of £10 regardless of how many
people attend – this is a group charge.
90
80
70
There are six marked points on the graph.
This graph is a linear graph, but it does
not show direct proportion because it
does not go through the origin.
Total cost (£)
60
50
40
30
Read up from 10 people on the x-axis to
the straight line. When you reach the line
move across horizontally until you reach
the y-axis. The cost is £30. This means
that 10 people will need to pay £30 to
attend the play.
20
10
0
0 10 20 30 40 50
Number of people attending
Distance–time graphs
Graphs that show the connection between the distance an object has
travelled and the time taken to travel that distance are called distance–time
graphs or travel graphs.
Time is normally shown along the horizontal axis and distance on the vertical.
The graphs normally start at the origin because at the beginning no time has
passed and no distance has been covered.
Look at the graph. It shows the following:
a cycle for 4 minutes from home
to a bus stop 1 km away
8
a 2 minute wait for the bus
7
a 7 km journey on the bus that
takes 10 minutes.
6
The line of the graph remains
horizontal while the person is
not moving (waiting for the bus)
because no distance is being
travelled at this time. The steeper
the line, the faster the person is
travelling.
580
Distance
travelled
(km)
5
4
3
2
1
0
0
2
4
6 8 10 12 14 16
Time (minutes)
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37 Interpreting graphs
Worked example 1
The graph shows the relationship between
the length and width of a hall.
Graph of length against width
10
Find the formula for this relationship.
9
8
7
6
Length (m) 5
4
3
2
1
0
(4, 10), (5, 8), (8, 5) and (10, 4)
4 3 10 5 40 m2
The area of the hall is constant, at 5 40 m2.
40   ​​ 
​length 5 ​ _____
width
0
1
2
3
4 5 6 7
Width (m)
8
9
10
Write down the coordinates of some points on the line.
Work out the area of the hall using one of the
coordinates, for example when the hall has a length of
10 m and a width of 4 m.
From the shape of the graph, you know that it is
showing inverse proportion, so as the length increases
the width decreases, therefore your formula will be
k
in the form y 5 ​​ __
x  ​​. k 5 40.
Because it shows a real-world context, the graph in Worked example 1 is only
valid for that particular range of values.
Tip
Graphs are also useful in the real world for reading off values quickly without
having to do the whole calculation. They can serve as conversion charts.
You saw conversion charts in
the form of exchange rates in
Chapter 34.
Worked example 2
This graph shows the number of Indian rupees you would get for
different numbers of US dollars at an exchange rate of US$1 : Rs 45.
This relationship is a direct proportion.
a Use the graph to estimate the dollar value of Rs 250.
b Use the graph to estimate how many rupees you could get for US$9.
500
400
Indian 300
rupees
(Rs) 200
100
a
Rs 250 is worth about $5.50.
b
You could get about Rs 400 for $9.
Find 250 on the y-axis and read
across and down to find the
corresponding point on the x-axis.
0
0
2
6
8
4
US dollars ($)
10
Find 9 on the x-axis and read up and across to
the find the corresponding point on the y-axis.
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
581
GCSE Mathematics for OCR (Foundation)
Exercise 37A
1 This graph shows the movement of a taxi during a four-hour period.
Movement of a taxi
10
8
Distance 6
(miles) 4
2
0
40
0
80
120
160
Time (minutes)
200
240
a Clearly and concisely describe the taxi’s journey.
b For how many minutes was the taxi waiting for passengers in this
period?
How can you tell this?
Tip
You saw in Chapter 14 that
distance travelled
  
  
speed 5 ​​  _________________
 ​​
time taken
c What was the total distance travelled?
d Calculate the taxi’s average speed during:
i the first 20 minutes.
ii the first hour.
iii from 160 to 210 minutes.
iv for the full period of the graph.
2 This distance–time graph represents Monica’s journey from home to a
supermarket and back again.
1200
Distance
from home
(metres)
800
400
0
09:00
09:10
09:20
Time
09:30
a How far was Monica from home at 09:06 hours?
b How many minutes did she spend at the supermarket?
c At what times was Monica 800 m from home?
d On which part of the journey did Monica travel faster, going to the
supermarket or returning home?
3 A swimming pool is 25 m long. Jasmine swims from one end to the
other in 20 seconds.
She rests for 10 seconds and then swims back to the starting point. It
takes her 30 seconds to swim the second length.
a Draw a distance–time graph for Jasmine’s swim.
b How far was Jasmine from her starting point after 12 seconds?
c How far was Jasmine from her starting point after 54 seconds?
582
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37 Interpreting graphs
4 A hurricane disaster centre has a certain amount of clean water. The
length of time the water will last depends on the number of people who
come to the centre.
a Calculate the missing values in this table.
No. of people
120 150 200 300 400
Days the water will last 40
32
b Plot a graph of this relationship.
Section 2: Gradients
Speed in distance–time graphs
Tip
The steepness (slope) or gradient of a distance–time graph gives an
indication of speed. A straight-line graph indicates a constant speed.
You learnt about kinematics in
Chapter 14. Revise that section if
you need to.
The steeper the graph, the greater the speed.
An upward slope and a downward slope represent movement in opposite
directions.
The distance–time graph shown is
for a person who walks, cycles and
then drives for three equal periods
of time.
15
For each period, speed is given by
the formula:
distance travelled
  
  
 ​​.
​speed 5 ​ _________________
time taken
If a line section on a graph is horizontal,
the gradient is zero and there is no speed,
that is, the object has stopped moving.
Distance
from start
(km)
10
drive
5
cycle
walk
0
0
10
20
Time (minutes)
30
Using gradient triangles to interpret changing gradients
Looking at the gradient of a graph along with the axis labels gives a large
amount of detail; even when there is no scale given.
This graph shows a car journey.
Distance
Time
Consider what happens as time moves on. In this case, as time moves on the
distance covered increases. So the car is moving.
Now consider the gradient triangles drawn on the graph. It doesn’t matter
where these triangles are drawn, each is similar to the other, so the sides
rise
represent the same gradient (​​ ​ ____
run ​ )​​.
This shows that the car is moving at a constant speed.
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
583
GCSE Mathematics for OCR (Foundation)
In this graph, as time moves on the distance
covered increases. So the car is moving.
Now consider the gradient triangles drawn on
the graph; each has the same base (unit of time)
but a different height.
Distance
This time the gradient triangles don’t fit the graph
as it is not a straight line. Instead, we’ve laid them
against the graph at different places. The
hypotenuse of each forms a tangent to the graph.
Time
You can see by the slope of each triangle’s hypotenuse that the speed is changing
along the graph. Moving up the slope, the triangles’ hypotenuses are getting steeper
– the gradient of the graph is increasing. This shows that the car is speeding up, or
accelerating.
Exercise 37B
1 The following graphs show what is happening to the level of water in a tank.
Describe what is happening in each case. Justify your answers using gradient
triangles.
a
b
Height
c
Height
Time
d
Height
Time
e
Time
f
Height
Height
Time
Height
Time
Time
2 The following graphs show what is happening to the price of oil.
Describe what is happening in each case, justifying your answers using
gradient triangles.
a
b
c
Price
Price
Time
d
Time
e
Price
Time
f
Price
Time
584
Price
Price
Time
Time
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37 Interpreting graphs
3 The following graph is a distance–time graph for a drag-racing car.
600
550
500
450
400
350
Distance
300
(m)
250
200
150
100
50
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Time (seconds)
a How far had the car travelled after 2 seconds?
b How long did it take the car to travel 50 metres?
c When was the car going at its fastest speed?
d How fast was the car going after
i 0.5 seconds?
ii 3.5 seconds?
4 The following graph shows the predicted height of the tide at Milford Haven.
Plot of the tidal heights predicted for Milford Haven – 7th November 2005
1.
00
2.
00
3.
00
4.
00
5.
00
6.
00
7.
00
8.
00
9.
0
10 0
.0
11 0
.0
12 0
.0
13 0
.0
14 0
.0
15 0
.0
16 0
.0
17 0
.0
18 0
.0
19 0
.0
20 0
.0
21 0
.0
22 0
.0
23 0
.0
0
7
6
5
Height above 4
chart datum 3
(metres)
2
1
0
Time
a When is the tide coming in at its fastest rate?
b When is the tide fully in?
c How fast is the tide going out at
i4pm?
ii2pm?
d Why would this kind of information be useful?
Checklist of learning and understanding
Graphs of real-world contexts
Graphs are useful for showing the relationships between quantities.
Graphs that show the connection between the distance an object has
travelled and the time taken are called distance–time graphs.
Find answers at: cambridge.org/ukschools/gcsemaths-studentbookanswers
585
GCSE Mathematics for OCR (Foundation)
Gradient
The gradient of a distance2time graph is the speed of the object.
○ If speed is constant the gradient is constant and is represented by a
straight line.
○ If the line is horizontal then the object is not moving; the gradient is zero.
Curved graphs have gradients that change along the graph continually.
Gradient triangles can be used to estimate the changes in the gradient.
or additional questions on
F
the topics in this chapter, visit
GCSE Mathematics Online.
Chapter review
1 The graph shows how the population of a village has changed since
1930.
300
200
Population
100
0
1930
1940
1950
1960
1970
Year
1980
1990
2000
2010
a Copy the graph using tracing paper and find the gradient of the graph
at the point (1950, 170).
b What does the gradient represent?
school
2 This graph shows Ben’s journey to school.
C
a During which part of his journey was
Ben travelling fastest?
Distance
b What happened between A and B?
A
B
c Did Ben speed up or slow down at C?
home
Time
3 The following graph shows the height of water in a cylindrical vase.
30
Height
(cm)
20
10
0
0
2
3
1
Time (minutes)
4
3
4
b Given that the radius of the base was 3 cm, what rate was the water
flowing at the beginning?
a The vase was filled to __
​​   ​​  of its capacity. How tall is the vase?
c What was the rate between 2 and 3 minutes?
586
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D