2

17. Characteristic Property of a Straight Line. Let P1 and P2 be any two points on a straight line, M1P1 and M2P2 their ordinates, and let PiQ be perpendicular to M2P2. Then Ax = PIQ is the difference of the abscissas of P1 and P2, and Ay QP2 is the difference of

the ordinates. The ratio of y1

the difference of the ordinates

P1

P

R

to that of the abscissas is

Ay QP2

Ax P1Q

For any second pair of

points on the line, P3 and P4,

the value of this ratio is

Since the triangles PiQP2 and P3RP are similar (why?),

The ratio of the difference of the ordinates of any two points on a straight line to the difference of the abscissas, Ay/Ax, is the same no matter what two points on the line be chosen.

This fact has been illustrated in the preceding section. Conversely, a line is a straight line if the ratio of ▲y/Ax is always the same for any two points on the line.

1

2

Let P1 and P2 be two definite points and På any third point on the given line which is to be proved straight. That the given line is straight will be proved by showing that P3 lies on the straight line determined by Pi and P2. Draw the line through P1 parallel to the x-axis cutting the ordinates of P2 and På at Q and R respectively. Since, by hypothesis, the values of ▲y/▲x computed for P1 and P2 and for P1 and P3 are equal, it follows that

QP 2 RP3

3

Hence the triangles PiQP2 and P1RP, are similar (why?),

3

and therefore ≤ QP1P2 = RP1Pз. Then PiP3 coincides with / Z 2 P1P2, and hence P3 lies on the straight line P1P2. The con

ขา

P2

P3

DEFINITION.

stant value of the ratio of the difference of the ordinates of any two points on a straight line to the difference of the abscissas is called the slope of the line.

If the slope is denoted by

m, this definition may be expressed in the symbolic form

Ay m = Ax

For any graph, the ratio of the difference of the ordinates to the difference of the abscissas, Ay/Ax, represents the average rate of change of a function. Hence the facts proved above may be stated as the

Theorem. If y is a function of x, the graph of y is a straight line if, and only if, the average rate of change of y with respect to x is constant.

To plot the graph of a function we usually express the functional relation in the form of an equation, build the table of values, and plot the curve. This process is unnecessary if it is known that the average rate of change of the function is constant, for the theorem just proved shows that the graph is a straight line. To draw the graph it is sufficient to obtain two pairs of values, plot the points representing them, and draw the straight line through these points.

EXAMPLE. Commercial alcohol, 95% pure, is poured into a bottle. Construct a graph showing the amount of alcohol in the bottle as a function of the amount of liquid in the bottle. Interpret the slope.

Plot the amount of liquid, l, on the horizontal axis, and the amount of alcohol, a, on the vertical axis. If Aa is the amount of alcohol in Al units of the liquid, we have ▲a = 0.95▲l, since 95% of the liquid is alcohol. Hence Aa/Al = 0.95, so that the rate of change of a with respect to l is

(10,9.5)

constant. The graph is therefore a straight line. If there is no liquid in the bottle, there is no alcohol, and hence the origin (0,0) is on the line; and if the bottle contains 10 units of liquid, there are 9.5 units of alcohol in the bottle. Hence (10, 9.5) is a second point on the line, and these two points are sufficient to determine the graph.

The slope represents the rate of change of a with respect to l, Aa/Al = 0.95, which gives the percentage of alcohol in the liquid.

·8

5

H

3

FIG. 31.

9 10 1

In plotting the graph by using the point (10, 9.5), it is assumed that the unit of volume is some such unit as an ounce or cubic centimeter. If a bottleful were chosen as the unit, which is convenient in many problems, it would be better to use the point (1, 0.95) and choose the unit on the l-axis as large as convenient. For then l 1 would mean that the bottle was full.

18. Slope of a Straight Line. "Slope of a line" (definition, Section 17) is the term used technically by mathematicians for what might be called the measure of steepness of the line. It may represent many things. Thus in the example in Section

16, As/At is the slope of the line, and hence the velocity v is represented by the slope.

If the slope m is computed from two points P1 (x1, y1) and P2(x2, y2) on the line, we have, in either figure,

In finding Ay and Ax it is essential that both coördinates of P1 be sub

tracted from those of P2, or vice versa.

Theorem 1. A line runs up to the right or down to the right according as its slope is positive or negative.

1

For two points P1 and P2 may be chosen on the line so that P1 lies to the left of P2, whence Ax is positive. Then Ay will be positive or negative according as P2 is above or below P1. Hence m = Ay/Ax will be positive or negative according as P2 is above or below P1, that is, according as the line runs up to the right or down to the right.

The precise relation between the value of m and the direction of the line as determined by the angle the line makes with the x-axis involves a transcendental function, and will be considered in a later chapter (see also Exercises 3 and 4 below).

The proof of the converse is left as an exercise.

Construction. To construct a line through a given point P1 whose slope is a given positive fraction a/b, take Q b units to the right of P1, and P2 a units above Q; then P1P2 is the required line. For we have

If the slope is negative, take P2 below Q. If the slope is an integer, it may be regarded as a fraction with unit denominator.

If two points are close together, the line through them cannot be drawn as accurately as if they were farther apart. Hence, in this construction, it is sometimes desirable to take PiQ equal to some multiple of b and QP2 equal to the same multiple of a.

EXERCISES

1. Plot the following pairs of points and the lines determined by them. Find ▲x, ▲y, and the slope m, and indicate the graphical significance of each of these quantities.

2. Construct the lines through the points indicated with the given slope.

(b) P(3, · 1), m =

(e) P(-2, 4), m =

(c) P(0, 4), m = 268 (f) P(5, 2), m = 0.

(a) P(2, 3), m 11/8 (d) P(3, 0), m = 2. 3. Show that of two lines through the same point the one which makes the greater acute angle with the x-axis has the larger slope numerically. 4. Show that the slope of a line parallel to the x-axis is zero.

5. If a telegram containing 10 words or less can be sent to a certain place for 35¢, and a 24-word message for 63¢, construct a graph showing the cost of a message containing any number of words, and determine the charge for each additional word above 10. What represents this charge?

6. Construct a graph to show the cost of any number of eggs at 40¢ a dozen. Determine from the graph the number of eggs which can be purchased for 70¢. What does the slope represent?

7. Construct a graph to show the amount of silver in any amount of an alloy containing 25% of silver. Determine from the graph how much silver there is in 20 pounds of the alloy. What does the slope represent?

8. A pound of an alloy contains 3 parts of silver and 5 of copper. Construct a graph to show the amount of silver in any amount of the alloy. Determine from the graph how much of the alloy contains 41 pounds of silver, and how much silver there is in 10 pounds of the alloy. Interpret the slope.

9. Ten ounces of alcohol 95% pure are poured into a bottle and then 5 ounces of water are added. Construct a graph showing the amount of alcohol in the bottle during the process as a function of the amount of liquid.

10. Solve Exercise 9 if 5 ounces of 50% alcohol are added instead of 5 ounces of water.

11. Is the slope of the line joining (3, 6) to (6, 12) the same as the slope of the line joining (3, 6) to ( – 3, -6)? What can be said of the three points?

12. Are the points (6, 1), (– 2, − 3), and (0, − 2) on a straight line? 13. Show that the points (2, 1), (3, 7), (5, 3), and (0, 5) are the vertices of a parallelogram.

14. Find the value of b if the line determined by (2, b) and (− 1, 3) is parallel to the line joining (– 4, – 5) and (7, − 2).