120. The Graph of an Equation. If $100 be invested at 3% simple interest, the interest amounts at the end of 1 year to $3, at the end of 2 years to $6, at the end of 3 years to $9, etc. This leads to the table below.

Figure 64 shows the graph drawn from this table as in § 118, with the years indicated on the horizontal line XX and the total interest on the vertical line YY.

There is another way of considering this matter. Since the interest amounts at the end of 1 year to $3, at the end of 2 years to $6, at the end of 3 years to $9, and so on, we see that if i stands for the interest at the end of t years, we have always the equation

i=3 t.

This equation gives in condensed form all that the above table and graph really say, for if we put t=1 in the equation we get (as we should) i=3, likewise, if we put t=2 in the equation it gives i=32, or 6, which agrees with the graph, etc. Thus, we have here

Y

2-1

20

1-9

1-8

1-7

1-6

1-51

1-3

1-2

1-1

10

-9

-8

7

-6

Interest (1 u

X

-41

3

5 6

Year

FIG. 64.

an equation and a graph which correspond to each other. Whenever this happens the graph is said to be the graph of the equation.

In such a case, the graph and the equation represent the same facts, but in two entirely different ways.

EXERCISES-GRAPHS OF EQUATIONS

1. Draw the graph like that of § 120, showing the interest on $100 at 2%, and write the corresponding equation.

2. Draw the graph like that of § 120, showing the interest on $100 at 4%, and write the corresponding equation.

3. Draw the graph of the equation i = 5 t.

[HINT.

First let t take the values 1, 2, 3, etc., and form a table. Then proceed as in § 120.]

4. A man leaves a certain place and walks at the rate of 2 miles per hour. Construct a graph to show how far he has walked after any given length of time, and find the corresponding equation.

[HINT. Let s stand for the distance walked in t hours, and see § 107.]

5. If a small heavy body, as a bullet, is dropped vertically downward from some high place, as a tower, the distance s (in feet) which it has passed over by the end of t seconds is determined by the equation s=16 t2. Draw the graph of this equation, letting a unit on the line XX represent 1 second, and a unit on the line YY represent 16 feet.

[HINT. The graph is not a straight line.]

The point

origin.

121. Definitions. Two lines perpendicular to each other (such as the lines XX and YY of § 117) are called a pair of axes. The horizontal line XX is called the axis of x, while the vertical line YY is called the axis of y. where the two lines meet (or cross) is called the Just as in Figure 18, page 34, we used the sign with numbers to the right of the origin (or zero point) and the sign with numbers to the left of that point, so we shall hereafter regard all distances on the x-axis (XX in Fig. 65)

asif measured to the right of the origin, and as if measured to the left of that point. Moreover, we shall carry out this same idea along the y-axis (YY in Fig. 65), thus considering as + all distances upward from the origin, and as all distances downward from that point.

X

+4

-41

4

2

133

Y

+4

P

+

X

0 1 2 3 4 5

Suppose we now think of a point which is not on either axis, as the point P in Fig. 65. The distance from P to the yaxis is seen to be 4 units (as measured along the x-axis below), and the distance from P to the x-axis is seen to be 3 units (as measured on the y-axis). More exactly, these measurements are +4 and +3 respectively, since the first is to the right, while the second is upward. In this way we may describe the location of P very briefly by calling it the point (+4,+3). In the same way, the point marked Q in the figure is the point (5, 4) because it is located 5 units to the left of the y-axis and 4 units upward. Again, R as it appears in the figure is the point (-4, -4) (Why?), while S is the point (6, 2) (Why?).

+6

R

41

=47

FIG. 65.

Observe carefully that the symbol for a point thus becomes in all cases (x, y) where x and y are numbers (positive or negative) the first of which gives the distance of the point from the. y-axis, while the second gives the distance from the x-axis. The numbers x and y are called the coördinates of the point; the first is sometimes called the "x of the point" and the second is called the "y of the point."

NOTE. If the point is on the x-axis its distance from that axis is O, hence such a point is represented by (x, 0). Similarly, a point on the y-axis becomes (0, y).

EXERCISES - COÖRDINATES

1. Locate on squared paper (plot) the points (-3, -1); (2, 4); (-1, 2); (3, 4); (-3, 7).

2. Plot the points (2, 3); (4, 5); (-3, -6); (−4, 2).

3. Plot the points (2, 1), (-8, 1), and (1, 6). Join these points by straight lines. What sort of figure is formed? Find the coördinates of the middle point of the horizontal side.

4. A certain street runs due east and west. It is met by another street which runs due north and south, thus forming a four corners." Taking the meeting point as origin and the east and north directions as the + XX and + YY axes, respectively, what are the coördinates of a flagpole which stands due southeast from the origin at a distance of 100 feet from each road? Answer the same question when the pole stands 100 feet due south of the crossing point.

5. Plot each of the following points and then see if it is possible to draw a straight line passing through all of them: (1, 2); (2, 4); (0, 0); (-1, -2); (-2, -4); (−3, −6); (-1, -1).

122. The Complete Graph of an Equation. The graph of an equation, as explained in § 120, becomes much more complete when we use a pair of axes such as we have in § 121. This will appear from the following example.

Example. Plot the graph of the equation 2 x+y=6.

Now give x any

SOLUTION. First draw a pair of axes. value in the equation; for example, let x = 1. The equation then becomes 2 × 1+ y = 6, or simply 2+ y = 6, from which, upon solving for y, we get y=4. Thus, the pair of values (x1,y=4) when used together satisfy the given equation. We now plot the point (1, 4) since it is what corresponds