The following theorems, whose proofs are left as exercises, follow from the facts that the points (x, y) and (x, y) are symmetrical to the point (x, y) with respect to the origin and the x-axis respectively.

Theorem 2A. The graph of f(x) is symmetrical with respect to the origin if

Theorem 2B. The graph of an equation is symmetrical with respect to the origin if the equation obtained by replacing x by —x and y by −y, and simplifying, is identical with the given equation.

-x

Theorem 3B. The graph of an equation is symmetrical with respect to the x-axis if the equation obtained by replacing y by -y, and simplifying, is identical with the given equation.

We shall use the phrase to discuss the table of values of a function to mean that the

Symmetry,

Values to be excluded,

Intercepts, and

Asymptotes (see next section)

are to be determined before building the table of values. For the last three considerations, solve the equation for y in terms of x and for x in terms of y. But if the intercepts are desired independently, they may be found by setting either variable equal to zero and solving for the other.

EXERCISES

1. Does f(x) always equal either ± f(x)?

2. Discuss the table of values (omitting asymptotes) and plot the graph of each of the functions and equations.

3. Discuss the table of values (omitting asymptotes) and plot the

graph of

y2 - 6x = 0. (i) y x4 - 4x2.

4. If f(x) is any one of the functions whose graphs are given below,

(a) x2 + y2

16.

(b)

x2 + y2 6x = 0.

(c) x2 + y2 + 4x 0.

(f) x2 - y2

f(x), find the value of ƒ(0), and the

yt

(c)

values of x for which f(x) is zero, and for which it is imaginary.

(ƒ)

11. Functions becoming Infinite. Asymptotes. Continuing the discussion in the preceding section, in the following example we shall need the

DEFINITION. It is said that a function becomes infinite as x approaches a if the numerical value of the function can be made larger than any positive number, however large, by giving x a value sufficiently near to a.

EXAMPLE. Build a table of values and plot the graph of the function

Symmetry. The tests for symmetry show that the graph is not symmetrical with respect to either axis or the origin. Solving (3) for x we get

Values excluded. As no radicals occur in (1) and (2), no imaginary values are encountered. Hence no values of x or y need be excluded on this account, and the graph runs indefinitely up and down, and to the right and left.

But the values x = 4 and y

0 must be excluded as it is impossible to divide by zero (see 3 (c), page xv).

Intercepts. Setting x = 0 in (1), the intercept on the y-axis is y 1. But as y 0 is an excluded value, the curve

does not cut the x-axis.

Asymptotes. To determine the form of the graph between the points corresponding to x = 3 and x = 5, the table includes a number of pairs of values near the excluded value x = 4.

1

x-4

1 1 1 2' 3' 4

X 0, 1, 2, 3, 3.5, 3.8, 3.9, 4, 4.1, 4.2, 4.5, 5, 6, 7, 8 -1 -1 -1 -1, -1, -2, -5, -10, ∞, 10, 5, 2, 1, 4' 3' 2' The numerical value of the function, according to the table, increases as x gets nearer to 4, and it is readily seen that by giving x values sufficiently near 4 the numerical value of

can be made larger than any given positive number,

however large. Thus to make

1

X 4

numerically greater than

10, let x have any value between 3.9 and 4.1 except the value 4; to make it greater than 100, let x have any value between 3.99 and 4.01 except the value 4; etc.

Hence the function

1

X 4

becomes infinite as x approaches 4.

This fact is indicated in the table by the symbols (4, ∞).

In the figure it is seen that as the graph gets nearer the line perpendicular to the x-axis at the point for which x = 4, it recedes farther and farther from the x-axis. Such a line is called an asymptote in accordance with the

DEFINITION. An asymptote of a curve is a straight line such that the distance from a point on the curve to the line approaches zero as the point recedes in

definitely from the origin along the curve.

Similar reasoning establishes the fact that the x-axis is also an asymptote, corresponding to the excluded value y = 0.

The example furnishes an illustration of the general principle:

If a function becomes infinite as x approaches a, the line perpendicular to the x-axis at x = a is an asymptote of the graph.

We shall consider asymptotes of the graph of an equation or

ม

function only when they are parallel to one of the coördinate axes. Asymptotes parallel to the y-axis may be found by solving the equation for y; the solution is a function of x, and to each value of x for which this function becomes infinite there corresponds a vertical asymptote. Horizontal asymptotes may be found in a similar manner by solving the equation for x.

An algebraic function becomes infinite for real, finite values of the variable only if the variable is contained in the denominator, and if the denominator is zero for one or more real values of the variable.

EXERCISES

1. Discuss the table of values and plot the graph of

2. If a gas is kept at the same temperature, the product of the pressure and the volume is constant. Discuss the table of values and plot the pressure as a function of the volume, assuming a numerical value for the constant.

12. Variation of a Function. Under this heading we consider primarily three things.

First. Sign of the function. If x increases through a given interval, the sign of the function will be positive if, and only if, the graph lies above the x-axis; and the sign will be negative if, and only if, the graph lies below the x-axis. For the ordinate representing a value of the function lies above or below the x-axis according as the function has a positive or negative value.

Second. Changes of the function. If x increases through a given interval, the value of the function increases if, and only if, the graph rises as it runs to the right; and the value decreases if, and only if, the graph falls as it runs to the right. For the ordinate representing a value of the function increases or decreases according as the curve rises or falls. The motion to the right corresponds to increasing values of x.

Third. Average rate of change of the function. This will be considered in the following section.

In order to state in what intervals the function is positive or negative, or is increasing or decreasing, it is necessary to determine the values of x bounding these intervals. These values of x, the corresponding values of the function, and the points on the graph representing them, are the remaining important elements of the variation of the function. They are:

First: The real zeros of the function, represented by the intercepts of the graph on the x-axis.

Second: The values of x for which the function becomes infinite, which give rise to the vertical asymptotes.

Third: The maximum and minimum values of the function, which we proceed to define.