In the previous section we saw some numeric examples on graphs. In this section we will see more examples. We will also discuss different types of graphs.
Solved example 2.4
Check whether the points (4,6), (6,7) and (10,9) lie on a line. If they do, find the coordinates of the point where the line intersect the Y axis.
Solution:
The points are plotted on a Graph sheet as shown in fig.2.13(a) below.
The 3 points lie on a line. The line is named as AB. We have to find the point where AB meets the Y axis. For that, we extend the line towards the left till it meets the Y axis. This is shown in the fig.2.13(b) below.
From the fig we can see that the point of intersection is (0,4)
If there are 2 given points, a line can be drawn between them. There is nothing special about it. But if a given third point also lies in that line, then there is much significance:
Solved example 2.4
Check whether the points (4,6), (6,7) and (10,9) lie on a line. If they do, find the coordinates of the point where the line intersect the Y axis.
Solution:
The points are plotted on a Graph sheet as shown in fig.2.13(a) below.
 |
| Fig.2.13(a) |
 |
| Fig.2.13(b) |
If there are 2 given points, a line can be drawn between them. There is nothing special about it. But if a given third point also lies in that line, then there is much significance:
If three given points lie on a line, then there is a definite relation between the three points. We will learn about it in later chapters. In the above example, we extended the line and found the point of intersection with the Y axis. This point of intersection is also another point on the line. The relation will hold good for that point also. In fact, the relation will hold good for all the points on the line.
Solved example 2.5
The x- coordinate of a point is 3. The distance of the point from the origin is 5. Find the y- coordinate.
Solution:
Let us draw a rough sketch. The details that we have are:
• The x- coordinate = 3. So the point lies some where on a line parallel to the Y axis, at a distance of 3 units from it.
• The distance from the origin is 5
The rough sketch will be as shown in fig.2.14(a) below:
From the rough sketch we can see that OPQ is a right angled triangle. PQ will be the required y- coordinate. Because PQ is the distance from X axis. To find PQ, we can use the pythagoras theorem. That is:
OQ2 + PQ2 = OP2 . So PQ = √(OP2 - OQ2) = √(52 - 32) = √(25 - 9) = √16 = 4. Thus we get y coordinate = 4. The final plot is shown in the fig.2.14(b)
We have seen a number of graphs so far in this chapter. In all of them, we referred the markings on the two axes as just 'units'. In the first example of a lamp, we used distances (in cm) on both the axes. In this way, we can use other quantities like time, temperature, speed, etc., on the axes. Also the quantities on both the axes need not be the same. On a same graph, one axis may show time and the other axis may show temperature. Let us see some examples of such graphs used in practice.
First we will see a Time – Distance graph. In this, time is plotted along one axis (usually the X axis) and distance along the other. We need coordinates of various points in the graph. Let us see how those coordinates are obtained:
We are going to plot the distance travelled by a car at various time points. We will need a stop watch for this. At the beginning, the time shown by the watch is zero. The car is at rest. The resting point of the car is marked by a point, say point 'O'.
Then the car begins to move. At that same instant, the watch is started. After five minutes, the distance travelled by the car is noted down. This can be obtained from the odometer in the car. After another five minutes the distance is again noted down. Note that 'another five minutes' means 10th from the beginning. The distance is noted down at the 15th, 20th, 25th and 30th minute. The readings can be tabulated as shown below:
With this table, we can plot a graph.
• Time is plotted along the X axis. So all the values in the first column of the table are x- coordinates
• Distance is plotted along the Y axis. So all the values in the second column of the table are the y- coordinates.
Thus the points are: O(0,0), A(5,3), B(10,6.2), C(15,8.2), D(20,11.7), E(25,15.3), F(30,18.1). The completed graph is shown in fig.2.15 below:
Let us see what all information we can obtain from the above graph:
The x- coordinates of each point is an increment of '5 mins' from it's previous point. So all the points are at a distance of '5 mins' apart along the Y axis. This is because we took readings every 5 minutes.
But the y- coordinates do not have such an uniformity. This means that
• the distance travelled in the second 5 mins is different from the distance travelled in the first five mins.
• the distance travelled in the third 5 mins is different from the distance travelled in the second five mins . . . and so on. Let us analyse this in more detail:
The total distance travelled up to A is 3 km (y- coordinate of A)
The total distance travelled up to B is 6.2 km (y- coordinate of B)
So the distance travelled from A to B is 6.2 – 3 =3.2 km
Similarly, the distance travelled from B to C = (y- coordinate of C) – (y- coordinate of B) = 8.2 – 6.2 = 2.0 km
So from B to C, the progress made by the car is less than that from A to B. There may be many reasons. May be the road from B to C lies with in the city centre, where there is much traffic congestion. Or may be, the road from B to C is in a very poor condition.
In the final laps, good progress is made: From D to E, the distance travelled is 15.3 – 11.7 = 3.6 km. And from E to F the distance travelled is 18.3 – 15.3 = 3.0 km. These regions may be away from the city center. Or, the roads may be well maintained.
Like this, a veriety of information can be obtained from such graphs. They find applications in fields such as New Town planning schemes, design of new roads and high ways, design and improvement of new cars and automobiles etc.,
In the next section we will see another Time - Distance graph.
Solved example 2.5
The x- coordinate of a point is 3. The distance of the point from the origin is 5. Find the y- coordinate.
Solution:
Let us draw a rough sketch. The details that we have are:
• The x- coordinate = 3. So the point lies some where on a line parallel to the Y axis, at a distance of 3 units from it.
• The distance from the origin is 5
The rough sketch will be as shown in fig.2.14(a) below:
 |
| Fig.2.14(a) |
From the rough sketch we can see that OPQ is a right angled triangle. PQ will be the required y- coordinate. Because PQ is the distance from X axis. To find PQ, we can use the pythagoras theorem. That is:
OQ2 + PQ2 = OP2 . So PQ = √(OP2 - OQ2) = √(52 - 32) = √(25 - 9) = √16 = 4. Thus we get y coordinate = 4. The final plot is shown in the fig.2.14(b)
 |
| Fig.2.14(b) |
We have seen a number of graphs so far in this chapter. In all of them, we referred the markings on the two axes as just 'units'. In the first example of a lamp, we used distances (in cm) on both the axes. In this way, we can use other quantities like time, temperature, speed, etc., on the axes. Also the quantities on both the axes need not be the same. On a same graph, one axis may show time and the other axis may show temperature. Let us see some examples of such graphs used in practice.
First we will see a Time – Distance graph. In this, time is plotted along one axis (usually the X axis) and distance along the other. We need coordinates of various points in the graph. Let us see how those coordinates are obtained:
We are going to plot the distance travelled by a car at various time points. We will need a stop watch for this. At the beginning, the time shown by the watch is zero. The car is at rest. The resting point of the car is marked by a point, say point 'O'.
Then the car begins to move. At that same instant, the watch is started. After five minutes, the distance travelled by the car is noted down. This can be obtained from the odometer in the car. After another five minutes the distance is again noted down. Note that 'another five minutes' means 10th from the beginning. The distance is noted down at the 15th, 20th, 25th and 30th minute. The readings can be tabulated as shown below:
| Time (min) | Distance (km) |
|---|---|
| 0 | 0 |
| 5 | 3 |
| 10 | 6.2 |
| 15 | 8.2 |
| 20 | 11.7 |
| 25 | 15.3 |
| 30 | 18.3 |
• Time is plotted along the X axis. So all the values in the first column of the table are x- coordinates
• Distance is plotted along the Y axis. So all the values in the second column of the table are the y- coordinates.
Thus the points are: O(0,0), A(5,3), B(10,6.2), C(15,8.2), D(20,11.7), E(25,15.3), F(30,18.1). The completed graph is shown in fig.2.15 below:
 |
| Fig.2.15 |
The x- coordinates of each point is an increment of '5 mins' from it's previous point. So all the points are at a distance of '5 mins' apart along the Y axis. This is because we took readings every 5 minutes.
But the y- coordinates do not have such an uniformity. This means that
• the distance travelled in the second 5 mins is different from the distance travelled in the first five mins.
• the distance travelled in the third 5 mins is different from the distance travelled in the second five mins . . . and so on. Let us analyse this in more detail:
The total distance travelled up to A is 3 km (y- coordinate of A)
The total distance travelled up to B is 6.2 km (y- coordinate of B)
So the distance travelled from A to B is 6.2 – 3 =3.2 km
Similarly, the distance travelled from B to C = (y- coordinate of C) – (y- coordinate of B) = 8.2 – 6.2 = 2.0 km
So from B to C, the progress made by the car is less than that from A to B. There may be many reasons. May be the road from B to C lies with in the city centre, where there is much traffic congestion. Or may be, the road from B to C is in a very poor condition.
In the final laps, good progress is made: From D to E, the distance travelled is 15.3 – 11.7 = 3.6 km. And from E to F the distance travelled is 18.3 – 15.3 = 3.0 km. These regions may be away from the city center. Or, the roads may be well maintained.
Like this, a veriety of information can be obtained from such graphs. They find applications in fields such as New Town planning schemes, design of new roads and high ways, design and improvement of new cars and automobiles etc.,
In the next section we will see another Time - Distance graph.
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