Graphing Essential Functions



Power functions

We inspected the behavior of power functions

f(x) = xn

for even and odd n.

We noted the relative positions of curves on their domain of R. On the interval (–1, 1) all power functions of this form are “flatter,” whether n is even or odd.

At (–1, 1) and (1, 1) the graphs of y = xn intersect.

And, on (–infinity,0) and (0, infinity) for n > m the graph y = xn is “steeper” than the graph of y = xm.

The notable difference in the power function graphs is seen in the comparison of even and odd-powered functions. The even powered functions form a U shape, whereas the odd powered functions are more like a snake.

We can picture this for n = 4 and m = 2, summarizing our observations as follows:

 


Likewise, for n = 3 and m =5,

 

By including a multiplier a, we get a “compression” of f(x) = axn as follows:

Example: f(x) = ax2
a = 2, 1, ½, –1

Example: f(x) = ax3
a = 2, 1, ½, –1

 

 

NOTE: If include add/subtract more x terms we get a polynomial of the form

This is not a simple power function. It often will no longer be an even or odd function. Instead, we have polynomial function that behaves according to a few other rules, which we look at sooner or later.















Rational functions

A rational function is a polynomial divided by a polynomial, although both are quite simple polynomials. The function

f(x) = 1/x

is an excellent starting point from which to build an understanding of rational functions in general. Just as we inspected power functions y = xn to begin the investigation of polynomial functions, for rational functions we look at the begin with the “mother function” f(x) = 1/x , one without a shift left or right, up or down.

Below is the graph of this function on [–4, 4]. Notice at x = 0 the function is not defined. And since we cannot solve y = 1/x = 0, then y will never be 0, either. However, the graph approaches both the axes and gets as close as it can without touching them. This is called asymptotic behavior.





Asymptotes and range:



Vertical asymptote: Clearly, the domain of f is



As x gets close to zero on both sides of the origin, the function rises steeply and heads for infinity when x is to the right of the y-axis (that is, x > 0), and it drops steeply and heads for negative infinity when x is to the left of the y-axis. A few values plotted on either side of x = 0 confirms this:

On the left of x = 0: f(–2) = –1/2 , f(–1) = –1 , f(–1/2) = –2 , f(–1/5) = –5, f(–1/100) = –100

On the right of x = 0: f(2) = 1/2 , f(1) = 1 , f(1/2) = 2 , f(1/5) = 5, f(1/100) = 100









We see the value of the functions becomes increasingly negative as x gets close to 0 (the asymptote) from the left, and it becomes increasingly positive as x gets close to 0 from the right.

But x can never = 0, so the graph never crosses the y-axis. For this reason, the y-axis is termed an asymptote. In fact, for any function of the form y = k/x the y-axis is the vertical asymptote.

Now look at the behavior of the function as x gets very large in either direction. (By “large” in the negative direction, we mean large absolute value.)

x gets large in the negative sense:

f(-10) = –1/10 , f(-100) = –1/100 , f(–1000) = –1/1000

x gets large in the positive sense:

f(10) = 1/10 , f(100) = 1/100 , f(1000) = 1/1000

We see the value of the function gets very close to zero coming from below zero as x gets very negative, and it gets very close to zero coming from above zero as x gets very positive.

But x can never = 0 (since 1/x can never equal zero), so the graph never crosses the y-axis. The x-axis functions as the horizontal asymptote for this function. As the function stretches to the left or right it gets closer and closer to the x-axis, but it never touches it. As x heads farther away from zero, in either the positive or negative direction, the function keeps the sign of x but gets smaller and smaller.

Thus, for the function f(x) = 1/x, the y-axis is a vertical asymptote, and the x-axis is a horizontal asymptote.

Any rigid transformation of a rational function will shift from the mother function pictured above to another set of asymptotes, above or below the x-axis, right or left of the y-axis.



Root functions

A root function is a function expressed by

f(x) = x1/n

for positive integer n greater than 1. So, it’s the inverse of the power function xn. Like power functions, the graphical representation of power functions is dependent upon whether n is even or odd.  

For even values of n (i.e., n = 2, 4, 6, ...), root functions will resemble the form illustrated for square root function expressed by f (x) = x1/2 depicted below. Its inverse is the function f (x) = x2 to its right if it were restricted to nonnegative values of x in its domain:

 


Clearly, root functions with even numbers for n only permit values of x ≥ 0, which is the domain. And just as the power function rises steeply, its inverse root function rises very slowly. Nevertheless, it attains as large a value of y as possible, so the range of the root function for n even is y ≥ 0.

Similarly, for odd values of n greater than 1 (i.e., n = 3, 5, 7, ...), root functions will resemble the form illustrated for cube root function expressed by f (x) = x1/3 depicted below, and its inverse power function f (x) = x3, complete with its original domain, is to its right:

It clear that the domain of the root function for odd n is all the reals. This must be true, as the range of the power function for odd n is all the reals, and the domain of an inverse function is the same as the range from which it is derived. The range of the root function for odd n is also all the reals, but the climb is very slow to the right, and the drop is very slow to the left.



Graphing power, root, and rational functions



Next we graph power functions that do not “sit” on (0, 0) (that is, they do not have symmetry across the y axis), root functions that do not originate at (0, 0), and rational functions that do not necessarily have x- and y-axes as their horizontal and vertical asymptotes.

These are functions that are have undergone transformations from their mother function, that is, they either shift without changing shape (so-called rigid transformations) or stretch or compress (so-called nonrigid transformations).