EXERCISES

1. Plot the points whose coördinates are given below, and determine the nature of the symmetry for the pairs of points in each group.

2. By means of the corresponding parts of Exercise 1, what can be said of the positions of the following pairs of points?

(a) (x, y) and (− x, y).

(c) (x, y) and (− x,

g).

(b) (x, y) and (x, − y).
(d) (x, y) and (y, x).

3. One end of a line bisected by the origin is the point (5, 2). What are the coördinates of the other end?

4. What are the coördinates of the point symmetrical to the point (-3, 4) with respect to the y-axis? the x-axis? the origin? the line bisecting the first and third quadrants?

5. Find the coördinates of the vertex or vertices not given in the regular polygons located as follows:

(a) One vertex of an equilateral triangle is the point (1, 0) and the altitude through this vertex, which is √3 units long, extends through the origin.

(b) An equilateral triangle has vertices with coördinates (0, 0) and (1, 0). (c) A square with opposite vertices having coördinates (1, 0) and (-1, 0).

(d) A hexagon two of whose opposite vertices have coördinates (1, 0) and (-1, 0).

(e) An octagon with two opposite vertices having coördinates (1, 0) and (-1, 0).

6. The coördinates of three vertices of a rhombus are (− 1, 0), (0, √3), (1, 0). What are the coördinates of the fourth vertex? (Three solutions.) What are the coördinates of the intersection of the diagonals?

9. Graph of a Function. Values of x and the corresponding values of a function may be exhibited in tabular form.

Table 1 gives the population of the United States in millions for the successive decades from 1830 to 1910.

Table 1.

x 1830, 1840, 1850, 1860, 1870, 1880, 1890, 1900, 1910, y 12.8, 17, 23.1, 31.4, 38.5, 50.1, 62.6, 75.9, 93.9,

Table 2 gives pairs of values of x and the function x + ·

Any such table of values may be strikingly exhibited to the eye by plotting the points whose coördinates are the pairs of numbers in the table, and then drawing a smooth curve through the points so obtained. Proceeding

are a pair of corresponding values of x and the function. Hence the important fact:

which have no graphs, and the graphs of others are merely one or more isolated points, but we shall not encounter them in this

course.

The graph of a function may be constructed by the following process:

Construct a table of values of x and the function of x.

Plot the points whose coördinates are the pairs of numbers in this table.

Draw a smooth curve through these points.

In constructing a graph, notice that values of x giving imaginary values of the function are discarded, and that the

number of points plotted must be large enough to indicate without doubt the form of the curve. Whenever it is not clear just how the curve is to be drawn, enlarge the table of values, either by giving more integral values of x, or by assuming for x intermediate values such as 2.5, 2.8, etc., as may best serve the purpose. The necessity for this last remark is shown by the fact that three points, situated as in Fig. 16(a), can be connected as in Figs. 16(b), (c), (d) for different types of function, and also in other ways (see Exercise 4 below).

DEFINITION. The graph of an equation in two variables is the curve such that: (1) Every point whose coördinates satisfy the equation is on the curve, and (2) Conversely, the coördinates of any point on the curve satisfy the equation.

To plot the graph of an equation,

Solve the equation for one of the variables in terms of the other, thus obtaining one as a function of the other.

Then proceed as indicated in the rule for the graph of a function. In many of the applications of the methods of coördinates, the coördinates refer to quantities of different kinds such as time, distance, work, cost, etc., and the graph represents a relation between two of these quantities.

EXERCISES

1. Construct the graphs of each of the following pairs of functions on the same axes. State a relation that each pair of graphs bear to one

another.

(a) 3x, 3x+2.

(b) - 2x, -2x+2. (c) x, x-3. (d) — 3x, – 3x – 2.

(e) −3x + 4, −3x – 5.

axes.

2. Construct the graphs of the following pairs of functions on the same Can the graph of one member of a set be moved so as to coincide with the graph of the other member? If so, how can this be done? (a) 2x2, 2x2 + 3. (b) x2 + 4x + 4, x2 + 4x. (c) — x2, - x2 +4. 3. Prove the Theorem: The graph of f(x) + k may be obtained by moving the graph of f(x) a distance k in the direction of the y-axis.

4. Plot the graphs of each of the functions in the following sets. To insure the proper connection of points obtained from integral values of x use intermediate values of x.

(a)

x3 - x2, 2x3 3x2+x, 2x3 x2 2.

(b) x3 + x2, − 2x3 + 3x2 − x, − x3 + 2x2

5. Plot the graphs of the following equations:

x.

(a) x

-3y+7 = 0.

(d) x2 + y2 = 36.

(b) y2 - 4x= 4.
(e) y + 2x2 - 4 = 0.

6. Distribution of the heights of 12-year-old boys. state one of its characteristics.

49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62.

} 1, 6, 18, 27, 45, 89, 115, 148, 123, 76, 34, 13, 5, 1.

of boys. NOTE. In constructing the graphs of some functions it may be desirable to choose different units of length on the x- and y-axes.

7. The following tables give the monthly receipts of eggs in the Chicago market in 1910, the price per dozen, and the storage of eggs by months in percentages of the total annual storage by a Chicago firm.

Month. Eggs in thousand cases.

Storage

of eggs

in percentages.

J. F. M. A. M. J. J. A.

S. O. N. D.

72, 140, 160, 760, 500, 400, 300, 240, 180, 120, 80, 48.

1.5, .8, 9.5, 42, 19, 22, 1, 0, 1, .3, .3, 2.6.

Price per 27, 27, 23, 21, 20, 18, 17, 20, 23, 26, 30, 33. dozen, cents.

Let time be the independent variable in each instance, and plot the three graphs on the same set of axes. Let a convenient length from the origin on the y-axis represent the three values, 800 thousand cases, 100%, and 40 cents a dozen, then mark off the subdivisions 50 thousand cases, 5%, and 5 cents a dozen, and multiples of these subdivisions.

State a relation that exists between each two of the three graphs.

What inferences can be drawn from the graphs with respect to the effect of storing eggs on the price?

8. The following are the monthly statistics for butter received, stored, and the price in the Chicago market in 1910.

Month.

Tubs re

J.

F. M. A. M. J. J. A. S. 0.

N. D.

ceived in 40, 52, 60, 72, 108, 168, 148, 104, 112, 96, 72, 56. thousands.

Storage in percentage.

Price per

lb. in cents.

} 6, 3.6, 2.8, 3.4, 14, 43, 10,

2, 4.2, 2.5, 4.5, 4.

34, 28, 31, 31, 26, 26, 26, 28, 27, 29, 30, 29.

Plot the three graphs on the same set of axes, as in problem 7. State a relation that exists between each two of the three functions. What inferences can be drawn from the graphs regarding the effect of storage of butter on the price?

NOTE. When a law is stated in such general terms that numerical data representing a concrete situation cannot be derived, a graph which will picture the general situation can be obtained by constructing a table of values with purely arbitrary sets of numbers which conform to the law. Such a graph will indicate the mode of change of the function with respect to the independent variable without representing a concrete situation.

9. The Weber-Fechner law of psychology states that as the intensity of an external stimulus increases in geometric progression, the corresponding sensation increases in arithmetic progression. Construct the graph. Let x represent the external stimuli, and y the corresponding sensations. Let the geometric progression 1, 2, 4, 8, be the values assumed for

x, and the arithmetic progression 1, 2, 3, 4, Plot the points whose coördinates are (1, the graph.

be the values assumed for y.

1), (2, 2), (4, 3), etc., and draw

At what value of x should the graph begin? Can there be an external stimulus without a corresponding sensation?

The law is said to hold for the senses of touch, hearing and seeing, but not for taste and smell.

10. Water pressure dies away uniformly because of the resistance of the conduits. Construct a graph to show the change in water pressure for points at different distances from a reservoir situated on a hill.

11. Malthus' law states that population increases in a geometric progression with reference to time, while subsistence increases in arithmetic progression. Plot and discuss the graphs with reference to the influence these two relations have on one another. The law of diminishing returns states that after a certain point, doubling the cultivation in agriculture will not double the returns. How will this affect the preceding graphs?