4. For a given abscissa, the ordinate of a point on the graph of y = f(x) + g(x) is the sum of the ordinates of the points with the same

FIG. 26.

abscissa on the graphs of f(x) and g(x). Hence the graph of y may be obtained by drawing the graphs of f(x) and g(x) and then adding the ordinates of points with the same abscissa. Using this method, which is called the addition of ordinates, construct the graphs of the following functions.

(a) Y

x2+x. Solution. Draw the graphs of x2 and of x. If MP and MQ are the ordinates of points on these graphs with the same abscissa OM, then MR MP + MQ = MP + PR is the ordinate of a point on the graph of the given function.

Several points on the graph may be obtained in this way, and a smooth curve drawn through them.

Notice that if OM is negative, so also is MQ, so that MR<MP.

(b) x2 + x.

(e) x3/3 + x/3.

(c) x2/3 + 2x.

(f) 1/x + x/2.

(d) x2/2+x. (g) x2 2x.

5. Express the area of a rectangle as a function of one of its sides, assuming that the perimeter is 8 feet. Plot the graph and discuss the variation of the function.

6. A house stands 50 feet from the street. A man is walking along the street. Express his distance from the house as a function of his distance from the entrance to the grounds, which is directly in front of the house. Discuss and plot the graph of the function.

7. A man 6 feet tall walks away from a lamp-post 12 feet high. Express the length of his shadow as a function of his distance from the post, and plot the graph of the function.

8. In Exercise 7, express the distance from the post to the shadow of the man's head as a function of the man's distance from the post. Plot the graph, and show that the average rate of change of the function is constant.

9. A point lies at a distance r from the origin. Find the equation expressing the functional relation between the coördinates x and y of the point.

10. A man walks for 2 hours at the rate of 3 miles an hour, stops an hour and a half for lunch, and walks back at the rate of 2 miles an hour. Construct a graph showing his distance from the starting point at any time.

1

NOTE. A solution of two simultaneous equations in x and y consists of a pair of values of these variables which satisfy both equations. If the numbers are real, the point whose coördinates are such a pair of values is on both graphs. Therefore a real solution of two simultaneous equations in x and y is represented by a point of intersection of the graphs. To find the coördinates of the points of intersection of two graphs, solve their equations simultaneously.

11. Plot the graphs of the following equations and find the coördinates of their points of intersection.

12. Plot the graphs of the following equations and from them read approximate values of the solutions of the equations.

13. Two engines in a freight yard are running on parallel tracks. Their distances from a signal tower (in hundred yards) at the time t (in minutes)

are given respectively by the equations

Plot the graphs, using t = 0, 4, 8, etc. Determine when and where the engines are beside each other (two solutions).

CHAPTER II

LINEAR FUNCTIONS

16. Uniform Rate of Change. The position of a body moving on a line, straight or curved, is commonly indicated by its distance s from a given point on the line. The body is said to move uniformly, or at a constant rate, along the line if the ratio of any change in s to the corresponding change in time is constant (that is, if this ratio has the same value for all intervals of time). If As is the change in s during the time At, then for uniform motion As/Atv, a constant called the velocity. The value of v gives the change in s during a unit of time.

Since As/At is the average rate of change of s during the interval of time At (see Section 13), the above definition is equivalent to the statement that a body moves uniformly at the rate of v units of distance per unit of time if the average rate of change of s is constant and always equal to v.

EXAMPLE. A farmer lives 10 miles from town. He starts from home and drives away from town at the uniform rate of 5 miles an hour. Construct and interpret a graph showling his distance from town at any time.

S

30

D

E

cording to a well-established custom. The pairs of values of t and s in the table are represented by the points A, B, C, D, E, which appear to lie on 46

:

a straight line, l. We assume that the graph is indeed straight, an assumption which will be justified in Section 20.

The interpretation of the coördinates of any point on the graph is that the ordinate represents the man's distance from town at the time represented by the abscissa.

The interpretation of the velocity is worthy of detailed consideration. The successive values of At given in the table are represented by AF

= BG = CH DI = 1, and those of As by FB = GC = HD = IE = 5. Hence v = As/At is represented by any one of the ratios

Now consider any interval of time beginning at t1 and ending at t2. The distances from town at these times are respectively, from (1),

with water, the rate at which a manufacturer turns out his product, etc.

A generalization of uniform rate of change with respect to time is given in the

DEFINITION. It is said that any variable y changes uniformly with respect to a second variable x, of which y is a function, if the ratio of any change in y to the change in x producing it, Ay/Ax, is equal to a constant, which is called the rate of change of y with respect to x. This rate gives the change in y due to a

unit change in x.

For example, if mercury is poured into a vertical tube, the weight of the mercury in the tube is a function of the height of the column of mercury. The ratio of any change in the weight to the change in height is constant, and equal to the weight of a column of unit height. Hence the weight changes uniformly with respect to the height.

It follows from the definition that if y changes uniformly with respect to x, equal changes in x produce equal changes in y.

EXERCISES

It is assumed that the graphs in the following exercises are straight lines. 1. Solve the example in Section 16 if the man starts toward town, considering the motion for 4 hours.

2. A through freight train was 90 miles from a city at 2 o'clock, and 150 miles at 4 o'clock. Construct the graph of its motion. Find its velocity, and interpret it geometrically. From the graph find how far the train was from the city at noon, and when it passed through the city. 3. A tank full of water is being emptied. After 5 minutes it contains 150 gallons, and after 12 minutes 108 gallons. Construct a graph showing the amount of water in the tank at any time. Find the capacity of the tank, the time required to empty it, and the rate at which it is being emptied. What represents each of these quantities graphically? 4. A man walks up a hill inclined at 30° to the horizon. Find the rate at which his altitude increases with respect to the distance he walks. Is it essential that he walk with a constant velocity?

5. A coal wagon is being filled with coal. Would the rate of change of the weight of the coal in the wagon with respect to the volume of the coal be constant? If so, how would it be measured?

6. Mention some quantity which is changing uniformly with respect to some other quantity, different from time, and state what the rate of change would be.

7. On the same axes, plot graphs showing the number of minute spaces the hands of a clock pass over in t minutes, starting at noon and ending at 1 o'clock. Determine from them how many spaces the hands are apart at 12:30.