## The Biology Project > Biomath > Power Function > Graphing

## Power Functions

Graphing Power Functions

In this section we will learn about the graphs of power functions,

f(x) = ax p.

To examine these graphs, we will begin by considering a = 1 and p ≥ 0. Notice that when
a
= 1 all such power functions go through the point (1, 1) since,

f(1) = (1) p = 1.

Graphing power functions where x > 0 and p ≥ 0

In many biological applications, we are concerned with positive values of x, so we will consider

f(x) = x p with x > 0 and p ≥ 0

by looking at several cases.

 Case 1: p = 0 The graph of the function is a straight line, the constant function f(x) = 1. Case 2: 0 < p < 1 The graph of the function is concave down and f(x) → ∞ as x → ∞. Case 3: p = 1 If p = 1, the power function reduces to the linear function f(x) = x. This case separates the behavior of f(x) = xp for 0 < p < 1 and p > 1. Case 4: p > 1 The graph of the function is concave up and f(x) → ∞ as x → ∞.

One important feature of power functions is how they compare to one another when
0 < x < 1 and x > 1. In particular, if 0 < x < 1, p > q implies xp < xq. For example, if 0 < x < 1,
then x2 < √x. On the other hand, if x > 1, p > q implies xp > xq . For instance, if x > 1, then x2 > √x. This feature of power functions can be seen in the plot below. Graphing power functions where p ≥ 0 and x < 0

What happens to the function f(x) = x p when p ≥ 0 and x < 0 is more complicated. \

• If p = r / s is a rational number expressed in lowest terms with s even or if p is an irrational number, f(x) = x p is not defined on the real line when x < 0.
• If p = r / s is a rational number expressed in lowest terms with s odd, f(x) = x p is defined for negative values of x.

The graph of f(x) when x < 0 will look one of two ways:

Case 1. If p = r / s (in lowest terms) with s odd and r even, f(x) → ∞ as x → − ∞, and the graph of f(x) is symmetric about the y-axis (i.e. f(x) is even). To see this, we interpret f(x) = xp as, and we show f(x) is even (with r even) as, The figure below depicts two such graphs. Case 2. If p = r / s (in lowest terms) with s odd and r odd, f(x) → −1 as x → −1, and the graph of f(x) is symmetric about the origin (i.e. f(x) is odd). To see this, we interpret f(x) = x p as, and we show f(x) is odd (with r odd) as, The figure below depicts two such graphs. Now we consider more complicated power functions where a ≠ 1 and p not necessarily greater than zero. The case a ≠ 1 can be handled by recalling graphical transformations. In particular, |a| > 1 vertically stretches the graph with respect to the base graph y = xp, while |a| < 1 vertically shrinks the graph with respect to the base graph. If a < 0 there is also a reflection about the x-axis.

Graphing power functions where p < 0.

The function f (x) = xp with p < 0 is not defined when x = 0 because division by zero is defined. Thus, we need to remove the point x = 0 from the domain since division by zero is undefined. If negative values of x are in the domain of f(x) = xp (i.e. if p = r / s in lowest terms with |s| odd), the behavior of f(x) near x = 0 increases or decreases without bound. In particular, the line x = 0 is a vertical asymptote of the graph. The graph of f(x) when p < 0 and f(x) defined for x < 0 will look one of two ways:

Case 1. If p = r / s < 0 (in lowest terms) with |s| odd and |r| even, f(x) → ∞ as x → 0- (i.e. as x approaches zero from the left), and f(x) → ∞ as x → 0 + (i.e. as x approaches 0 from the right). The figure below depicts this behavior Case 2. If p = r / s < 0 (in lowest terms) with |s| odd and |r| odd, f(x) → − ∞ as x → 0- and f(x) → ∞ as x → 0+. The figure below depicts this behavior. If negative real numbers are not in the domain of f (x) = xp with p < 0 (i.e. if p = r/s in lowest terms with |s| even or p is an irrational number), then f (x) → ∞ as x → 0+. The graph of f(x) will look lie the figure below. *****

Now try some problems that will test your understanding of power functions.

Problems

The Biology Project > Biomath > Power Functions > Graphing

The Biology Project
Department of Biochemistry and Molecular Biophysics
The University of Arizona

April 2006
Contact the Development Team