NCERT Textbook- Introduction to Graphs | NCERT Textbooks (Class 6 to Class 12) - CTET & State TET PDF Download

 Page 1


INTRODUCTION TO GRAPHS  157
13.1  Introduction
Have you seen graphs in the newspapers, television, magazines, books etc.? The purpose
of the graph is to show numerical facts in visual form so that they can be understood
quickly, easily and clearly. Thus graphs are visual representations of data collected.
Data can also be presented in the form of a table; however a graphical presentation is
easier to understand. This is true in particular when there is a trend or comparison to
be shown.
We have already seen some types of graphs. Let us quickly recall them here.
13.1.1  A line graph
A line graph displays data that changes continuously over periods of time.
When Renu fell sick, her doctor maintained a record of her body temperature, taken
every four hours. It was in the form of a graph (shown in Fig 13.1 and Fig 13.2).
W e may call this a “time-temperature graph”.
It is a pictorial representation of the following data, given in tabular form.
Time 6 a.m. 10 a.m. 2 p.m. 6 p.m.
Temperature(°C) 37 40 38 35
The horizontal line (usually called the x-axis) shows the timings at which the
temperatures were recorded. What are labelled on the vertical line (usually called
the y-axis)?
Introduction to Graphs
CHAPTER
1 3
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INTRODUCTION TO GRAPHS  157
13.1  Introduction
Have you seen graphs in the newspapers, television, magazines, books etc.? The purpose
of the graph is to show numerical facts in visual form so that they can be understood
quickly, easily and clearly. Thus graphs are visual representations of data collected.
Data can also be presented in the form of a table; however a graphical presentation is
easier to understand. This is true in particular when there is a trend or comparison to
be shown.
We have already seen some types of graphs. Let us quickly recall them here.
13.1.1  A line graph
A line graph displays data that changes continuously over periods of time.
When Renu fell sick, her doctor maintained a record of her body temperature, taken
every four hours. It was in the form of a graph (shown in Fig 13.1 and Fig 13.2).
W e may call this a “time-temperature graph”.
It is a pictorial representation of the following data, given in tabular form.
Time 6 a.m. 10 a.m. 2 p.m. 6 p.m.
Temperature(°C) 37 40 38 35
The horizontal line (usually called the x-axis) shows the timings at which the
temperatures were recorded. What are labelled on the vertical line (usually called
the y-axis)?
Introduction to Graphs
CHAPTER
1 3
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158  MATHEMATICS
       Fig 13.1                           Fig 13.2
Each piece of data is shown The points are then connected by line
by a point on the square grid. segments. The result is the line graph.
What all does this graph tell you? For example you can see the pattern of temperature;
more at 10 a.m. (see Fig 13.3) and then decreasing till 6 p.m. Notice that the temperature
increased by 3° C(= 40° C – 37° C) during the period 6 a.m. to 10 a.m.
There was no recording of temperature at 8 a.m., however the graph suggests that it
was more than 37 °C (How?).
Example 1: (A graph on “performance”)
The given graph (Fig 13.3) represents the total runs scored by two batsmen A and B,
during each of the ten different matches in the year 2007. Study the graph and answer the
following questions.
(i) What information is given on the two axes?
(ii) Which line shows the runs scored by batsman A?
(iii) Were the run scored by them same in any match in 2007? If so, in which match?
(iii) Among the two batsmen, who is steadier? How do you judge it?
Solution:
(i) The horizontal axis (or the x-axis) indicates the matches played during the year
2007. The vertical axis (or the y-axis) shows the total runs scored in each match.
(ii) The dotted line shows the runs scored by Batsman A. (This is already indicated at
the top of the graph).
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INTRODUCTION TO GRAPHS  157
13.1  Introduction
Have you seen graphs in the newspapers, television, magazines, books etc.? The purpose
of the graph is to show numerical facts in visual form so that they can be understood
quickly, easily and clearly. Thus graphs are visual representations of data collected.
Data can also be presented in the form of a table; however a graphical presentation is
easier to understand. This is true in particular when there is a trend or comparison to
be shown.
We have already seen some types of graphs. Let us quickly recall them here.
13.1.1  A line graph
A line graph displays data that changes continuously over periods of time.
When Renu fell sick, her doctor maintained a record of her body temperature, taken
every four hours. It was in the form of a graph (shown in Fig 13.1 and Fig 13.2).
W e may call this a “time-temperature graph”.
It is a pictorial representation of the following data, given in tabular form.
Time 6 a.m. 10 a.m. 2 p.m. 6 p.m.
Temperature(°C) 37 40 38 35
The horizontal line (usually called the x-axis) shows the timings at which the
temperatures were recorded. What are labelled on the vertical line (usually called
the y-axis)?
Introduction to Graphs
CHAPTER
1 3
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158  MATHEMATICS
       Fig 13.1                           Fig 13.2
Each piece of data is shown The points are then connected by line
by a point on the square grid. segments. The result is the line graph.
What all does this graph tell you? For example you can see the pattern of temperature;
more at 10 a.m. (see Fig 13.3) and then decreasing till 6 p.m. Notice that the temperature
increased by 3° C(= 40° C – 37° C) during the period 6 a.m. to 10 a.m.
There was no recording of temperature at 8 a.m., however the graph suggests that it
was more than 37 °C (How?).
Example 1: (A graph on “performance”)
The given graph (Fig 13.3) represents the total runs scored by two batsmen A and B,
during each of the ten different matches in the year 2007. Study the graph and answer the
following questions.
(i) What information is given on the two axes?
(ii) Which line shows the runs scored by batsman A?
(iii) Were the run scored by them same in any match in 2007? If so, in which match?
(iii) Among the two batsmen, who is steadier? How do you judge it?
Solution:
(i) The horizontal axis (or the x-axis) indicates the matches played during the year
2007. The vertical axis (or the y-axis) shows the total runs scored in each match.
(ii) The dotted line shows the runs scored by Batsman A. (This is already indicated at
the top of the graph).
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INTRODUCTION TO GRAPHS  159
(iii) During the 4th match, both have scored the same
number of 60 runs. (This is indicated by the point
at which both graphs meet).
(iv) Batsman A has one great “peak” but many deep
“valleys”. He does not appear to be  consistent.
B, on the other hand has never scored below a
total of 40 runs, even though his highest score is
only 100 in comparison to 115 of A. Also A has
scored a zero in two matches and in a total of 5
matches he has scored less than 40 runs. Since A
has a lot of ups and downs, B is a more consistent
and reliable batsman.
Example 2:  The given graph (Fig 13.4) describes
the distances of a car from a city P at different times
when it is travelling from City P to City Q, which are
350 km apart. Study the graph and answer the following:
(i) What information is given on the two axes?
(ii) From where and when did the car begin its
journey?
(iii) How far did the car go in the first hour?
(iv) How far did the car go during (i) the 2nd hour? (ii) the 3rd hour?
(v) Was the speed same during the first three hours? How do you know it?
(vi) Did the car stop for some duration at any place? Justify your answer.
(vii) When did the car reach City Q?
Fig 13.3
Fig 13.4
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INTRODUCTION TO GRAPHS  157
13.1  Introduction
Have you seen graphs in the newspapers, television, magazines, books etc.? The purpose
of the graph is to show numerical facts in visual form so that they can be understood
quickly, easily and clearly. Thus graphs are visual representations of data collected.
Data can also be presented in the form of a table; however a graphical presentation is
easier to understand. This is true in particular when there is a trend or comparison to
be shown.
We have already seen some types of graphs. Let us quickly recall them here.
13.1.1  A line graph
A line graph displays data that changes continuously over periods of time.
When Renu fell sick, her doctor maintained a record of her body temperature, taken
every four hours. It was in the form of a graph (shown in Fig 13.1 and Fig 13.2).
W e may call this a “time-temperature graph”.
It is a pictorial representation of the following data, given in tabular form.
Time 6 a.m. 10 a.m. 2 p.m. 6 p.m.
Temperature(°C) 37 40 38 35
The horizontal line (usually called the x-axis) shows the timings at which the
temperatures were recorded. What are labelled on the vertical line (usually called
the y-axis)?
Introduction to Graphs
CHAPTER
1 3
Reprint 2024-25
158  MATHEMATICS
       Fig 13.1                           Fig 13.2
Each piece of data is shown The points are then connected by line
by a point on the square grid. segments. The result is the line graph.
What all does this graph tell you? For example you can see the pattern of temperature;
more at 10 a.m. (see Fig 13.3) and then decreasing till 6 p.m. Notice that the temperature
increased by 3° C(= 40° C – 37° C) during the period 6 a.m. to 10 a.m.
There was no recording of temperature at 8 a.m., however the graph suggests that it
was more than 37 °C (How?).
Example 1: (A graph on “performance”)
The given graph (Fig 13.3) represents the total runs scored by two batsmen A and B,
during each of the ten different matches in the year 2007. Study the graph and answer the
following questions.
(i) What information is given on the two axes?
(ii) Which line shows the runs scored by batsman A?
(iii) Were the run scored by them same in any match in 2007? If so, in which match?
(iii) Among the two batsmen, who is steadier? How do you judge it?
Solution:
(i) The horizontal axis (or the x-axis) indicates the matches played during the year
2007. The vertical axis (or the y-axis) shows the total runs scored in each match.
(ii) The dotted line shows the runs scored by Batsman A. (This is already indicated at
the top of the graph).
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INTRODUCTION TO GRAPHS  159
(iii) During the 4th match, both have scored the same
number of 60 runs. (This is indicated by the point
at which both graphs meet).
(iv) Batsman A has one great “peak” but many deep
“valleys”. He does not appear to be  consistent.
B, on the other hand has never scored below a
total of 40 runs, even though his highest score is
only 100 in comparison to 115 of A. Also A has
scored a zero in two matches and in a total of 5
matches he has scored less than 40 runs. Since A
has a lot of ups and downs, B is a more consistent
and reliable batsman.
Example 2:  The given graph (Fig 13.4) describes
the distances of a car from a city P at different times
when it is travelling from City P to City Q, which are
350 km apart. Study the graph and answer the following:
(i) What information is given on the two axes?
(ii) From where and when did the car begin its
journey?
(iii) How far did the car go in the first hour?
(iv) How far did the car go during (i) the 2nd hour? (ii) the 3rd hour?
(v) Was the speed same during the first three hours? How do you know it?
(vi) Did the car stop for some duration at any place? Justify your answer.
(vii) When did the car reach City Q?
Fig 13.3
Fig 13.4
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160  MATHEMATICS
Solution:
(i) The horizontal (x) axis shows the time. The vertical (y) axis shows the distance of the
car from City P .
(ii) The car started from City P at 8 a.m.
(iii) The car travelled 50 km during the first hour. [This can be seen as follows.
At 8 a.m. it just started from City P . At 9 a.m. it was at the 50th km (seen from graph).
Hence during the one-hour time between 8 a.m. and 9 a.m. the car travelled 50 km].
(iv) The distance covered by the car during
(a) the 2nd hour (i.e., from 9 am to 10 am) is 100 km, (150 – 50).
(b) the 3rd hour (i.e., from 10 am to 11 am) is 50 km (200 – 150).
(v) From the answers to questions (iii) and (iv), we find that the speed of the car was not
the same all the time. (In fact the graph illustrates how the speed varied).
(vi) We find that the car was 200 km away from city P when the time was 11 a.m. and
also at 12 noon. This shows that the car did not travel during the interval 11 a.m. to
12 noon. The horizontal line segment representing “travel” during this period is
illustrative of this fact.
(vii) The car reached City Q at 2 p.m.
EXERCISE 13.1
1. The following graph shows the temperature of a patient in a hospital, recorded
every hour.
(a) What was the patient’s temperature at 1 p.m. ?
(b) When was the patient’s temperature 38.5° C?
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INTRODUCTION TO GRAPHS  157
13.1  Introduction
Have you seen graphs in the newspapers, television, magazines, books etc.? The purpose
of the graph is to show numerical facts in visual form so that they can be understood
quickly, easily and clearly. Thus graphs are visual representations of data collected.
Data can also be presented in the form of a table; however a graphical presentation is
easier to understand. This is true in particular when there is a trend or comparison to
be shown.
We have already seen some types of graphs. Let us quickly recall them here.
13.1.1  A line graph
A line graph displays data that changes continuously over periods of time.
When Renu fell sick, her doctor maintained a record of her body temperature, taken
every four hours. It was in the form of a graph (shown in Fig 13.1 and Fig 13.2).
W e may call this a “time-temperature graph”.
It is a pictorial representation of the following data, given in tabular form.
Time 6 a.m. 10 a.m. 2 p.m. 6 p.m.
Temperature(°C) 37 40 38 35
The horizontal line (usually called the x-axis) shows the timings at which the
temperatures were recorded. What are labelled on the vertical line (usually called
the y-axis)?
Introduction to Graphs
CHAPTER
1 3
Reprint 2024-25
158  MATHEMATICS
       Fig 13.1                           Fig 13.2
Each piece of data is shown The points are then connected by line
by a point on the square grid. segments. The result is the line graph.
What all does this graph tell you? For example you can see the pattern of temperature;
more at 10 a.m. (see Fig 13.3) and then decreasing till 6 p.m. Notice that the temperature
increased by 3° C(= 40° C – 37° C) during the period 6 a.m. to 10 a.m.
There was no recording of temperature at 8 a.m., however the graph suggests that it
was more than 37 °C (How?).
Example 1: (A graph on “performance”)
The given graph (Fig 13.3) represents the total runs scored by two batsmen A and B,
during each of the ten different matches in the year 2007. Study the graph and answer the
following questions.
(i) What information is given on the two axes?
(ii) Which line shows the runs scored by batsman A?
(iii) Were the run scored by them same in any match in 2007? If so, in which match?
(iii) Among the two batsmen, who is steadier? How do you judge it?
Solution:
(i) The horizontal axis (or the x-axis) indicates the matches played during the year
2007. The vertical axis (or the y-axis) shows the total runs scored in each match.
(ii) The dotted line shows the runs scored by Batsman A. (This is already indicated at
the top of the graph).
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INTRODUCTION TO GRAPHS  159
(iii) During the 4th match, both have scored the same
number of 60 runs. (This is indicated by the point
at which both graphs meet).
(iv) Batsman A has one great “peak” but many deep
“valleys”. He does not appear to be  consistent.
B, on the other hand has never scored below a
total of 40 runs, even though his highest score is
only 100 in comparison to 115 of A. Also A has
scored a zero in two matches and in a total of 5
matches he has scored less than 40 runs. Since A
has a lot of ups and downs, B is a more consistent
and reliable batsman.
Example 2:  The given graph (Fig 13.4) describes
the distances of a car from a city P at different times
when it is travelling from City P to City Q, which are
350 km apart. Study the graph and answer the following:
(i) What information is given on the two axes?
(ii) From where and when did the car begin its
journey?
(iii) How far did the car go in the first hour?
(iv) How far did the car go during (i) the 2nd hour? (ii) the 3rd hour?
(v) Was the speed same during the first three hours? How do you know it?
(vi) Did the car stop for some duration at any place? Justify your answer.
(vii) When did the car reach City Q?
Fig 13.3
Fig 13.4
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160  MATHEMATICS
Solution:
(i) The horizontal (x) axis shows the time. The vertical (y) axis shows the distance of the
car from City P .
(ii) The car started from City P at 8 a.m.
(iii) The car travelled 50 km during the first hour. [This can be seen as follows.
At 8 a.m. it just started from City P . At 9 a.m. it was at the 50th km (seen from graph).
Hence during the one-hour time between 8 a.m. and 9 a.m. the car travelled 50 km].
(iv) The distance covered by the car during
(a) the 2nd hour (i.e., from 9 am to 10 am) is 100 km, (150 – 50).
(b) the 3rd hour (i.e., from 10 am to 11 am) is 50 km (200 – 150).
(v) From the answers to questions (iii) and (iv), we find that the speed of the car was not
the same all the time. (In fact the graph illustrates how the speed varied).
(vi) We find that the car was 200 km away from city P when the time was 11 a.m. and
also at 12 noon. This shows that the car did not travel during the interval 11 a.m. to
12 noon. The horizontal line segment representing “travel” during this period is
illustrative of this fact.
(vii) The car reached City Q at 2 p.m.
EXERCISE 13.1
1. The following graph shows the temperature of a patient in a hospital, recorded
every hour.
(a) What was the patient’s temperature at 1 p.m. ?
(b) When was the patient’s temperature 38.5° C?
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INTRODUCTION TO GRAPHS  161
(c) The patient’s temperature was the same two times during the period given.
What were these two times?
(d) What was the temperature at 1.30 p.m.? How did you arrive at your answer?
(e) During which periods did the patients’ temperature showed an upward trend?
2. The following line graph shows the yearly sales figures for a manufacturing company .
(a) What were the sales in (i) 2002     (ii) 2006?
(b) What were the sales in (i) 2003     (ii) 2005?
(c) Compute the difference between the sales in 2002 and 2006.
(d) In which year was there the greatest difference between the sales as compared
to its previous year?
3. For an experiment in Botany, two different plants, plant A and plant B were grown
under similar laboratory conditions. Their heights were measured at the end of each
week for 3 weeks. The results are shown by the following graph.
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