n>1. In this case the graph of

y

= xn

(1)

is tangent to the x-axis at the origin (0, 0), rises to the right and passes through the point (1, 1), at which the slope of the tangent line is n. That it passes through these points is seen by substituting their coördinates in (1).

To find the slope of the line tangent to the graph at any point, when n is a positive integer, replace x by x + Ax and y by y + Ay in (1), which gives y + ▲y = (x + Ax)”.

Expanding the right hand member by the binomial theorem,

Passing to the limit as Ax approaches zero, the slope of the

If n is a positive integer greater than unity, the part of the graph of " in the first quadrant is then very much like the graph of x2.

If n is a positive fraction greater than unity the part of the graph in the first quadrant has the same general appearance. To see this, let r, s, and t be three values of n such that r<s<t. For a positive value of x the value of 2 will lie between x and x2, and hence the part of the graph of x in the first quadrant lies between the graphs of x and xt (this holds for all values of r, s, and t). For example, the graph of x lies between the graphs of x and x2; that of x between those of x2 and x3; etc. The figures show the graphs of x2, x3, and x, which are symmetrical with respect to the y-axis, the origin, and the x-axis

respectively. The symmetry of the last curve is seen by writing y = x in the form y2 = 23, and applying Theorem 3B, page 24. The graph of x", n>1, always resembles one of these curves. Thus the graph of x4 is very much like a parabola, but it is flatter near the origin and steeper elsewhere.

The x-axis is tangent to the graph of x3 at the origin. This tangent differs from any we have encountered hitherto in that it crosses the curve at the point of tangency.

The graph of x is remarkable in that it has a sharp point at the origin. It differs from other curves we have studied in detail in that vertical lines to the right of the y-axis cut it

==

twice, corresponding to the fact that x = V has two values for each positive value of x.

=

y2.

ур S

Q (y, x)

39. Graph of x", 0<n<1. Graphs of Inverse Functions. Let us first consider n = 1, or the function x. If we set y = x, and solve for x, we get x This differs from the equation y = x2 only in that x and y have been interchanged. Hence if (x, y) is a point on the graph of either equation, the point (y, x) is on the graph of the other. But these points are symmetrical with respect to the bisector of the first and third quadrants, as may be established from the figure. Hence the graphs

N

P (x,y)

T

Ꭱ

Mx

FIG. 59.

of the two equations, that is, the graphs of x and x2, are symmetrical to each other with respect to this bisector.

The graph of x may therefore be constructed as follows: Construct the graph of x2 and the bisector of the first and

y

2

6

5

-2

5

FIG. 60.

6 x

third quadrants. Choose a number of points on the graph of x2, and construct the points symmetrical to them with respect to this bisector. Draw a smooth curve through the points so obtained.

We may now get properties of the function by interpreting its graph. For example, since the graph of x2 is symmetrical with respect to the y-axis that of x is symmetrical with respect to the x-axis, and hence to each value of x there correspond two values of x1 which are equal numerically but differ in sign. And since no part of the graph lies to the left of the y-axis (why?), the function is imaginary if x is negative. What other properties may be obtained in this way?

If two curves are symmetrical to each other with respect to a line they are congruent, for one may be brought into co

incidence with the other by rotating the plane about the line through 180°. Hence the graph of x is a parabola.

Since we obtained the equation y = x2 by solving y = x1 for x and then interchanging x and y, the functions x2 and x are inverse functions (page 40). The graphical considerations above may be applied to any two inverse functions, and hence we have the

Theorem. The graphs of inverse functions are symmetrical to each other with respect to the bisector of the first and third quadrants.

If the graph of any function is given, then the graph of the inverse function may be obtained readily by this theorem. The distinction between two curves symmetrical to each other with respect to a line and a single curve which is symmetrical with respect to a line should be noted.

To find the inverse of x", set y=x", and interchange x and y which gives x = y"; solving for y we get y=x. Hence the inverse of x" is x, and the graphs of these functions are

symmetrical to each other with respect to the bisector of the first and third quadrants.

By means of this symmetry, for example, the graph of t

Colto

may be obtained from that of x3, and the graph of x3 from that of x.

The graph of x2, if 0<n<1, always resembles the graph of one of the functions x, x, and x. In the first quadrant, the graph is tangent to the y-axis at the origin, rises to the right, and passes through the point (1, 1), at which the slope of the tangent line is n. Hence the smaller the value of n, the less rapidly the graph rises at this point, from which it follows that it must be steeper near the origin. The remaining part of the graph may be determined by means of the symmetry of the curve with respect to one of the axes or the origin.

EXERCISES

1. Plot the graph of x" for each of the values of n given below, using the same axes.

(a) n =.2, 4, 6. What can be said of the graph if n is an even positive integer?

(b) n = 3, 5. What can be said of the graph when n is an odd positive integer?

for n =

(c) n = 2. Find the slope of the tangent line at any point to the graph of x" 2, 3, 4, 5. Find and tabulate the slope for each graph at the points for which x = 0, 0.3, 0.5, 0.8, 1, 2. As n increases, what can be said of the slope of the tangent line at a point near the origin? Remote from the origin?

1, 2, 3, 1, 1, 1, 3, using as large a scale as possible.

3. On separate axes sketch the graphs of TM" for n = 号,爭,秀 Is there much difference between the parts of these curves in the first quadrants? 4. Find the inverse of each of the functions in Exercise 3, and sketch their graphs.

5. Determine the symmetry of x" when n is a fraction p/q, if (a) p is even and q is odd; (b) p is odd and q is even; (c) p and q are both odd. What values of n give typical forms of the graph in these three cases if p/q>1? If p/q<1?

6. Plot the graph of 24, find the inverse function, and plot its graph. 7. Would it be easy to build a table of values for xt? Construct its graph by first finding the inverse function.

8. Plot the graphs of the following functions, in each case finding the inverse function and its graph.

(a) y = x2 - 2.

(c) y =

(b) y

=

x2 + 2x.

0.2, 0.4, 0.6,

− x2 + 4x − 4.

9. Using Huntington's Tables, construct the graphs, on the same axes, of x, x2, x3, x2, x3, taking x = 1.4. Use as large a scale as possible and plot the parts of the graphs in the first quadrant only.