Power Functions: Case 1: a = n (n = 1,2,3...)

In the next few posts, we will have an in-depth look at power functions. These are functions where the dependent variable x is raised to the power a. That is, they are of the form...

Case 1. The power a = n, where n is a positive integer. That is...

These are polynomial functions of degree n with only one term. Graphs with increasing powers are shown in the tables below...

Notice that when the powers are odd integers, we have odd functions, and we have an inflection point at the origin (0,0). When the powers are even integers, we have even functions, and a turning point at the origin.

With the exception of n = 1, graphs with odd powers look like x³, but they get progressively flatter when x is near 0. This is because for small values of x; x³, x⁵ and so on, become smaller and smaller.

They get progressively steeper when |x| ≥ 1, because for larger values of x; x³, x⁵ and so on, become larger and larger.

The same applies to even powers, except they look similar to x² (a parabola).

The power functions of odd and even integers are overlaid on top of each other in Figures 1 and 2 respectively.

Figure 1.

Figure 2.

Notice that for the odd powers, they have common intersection points at (-1,-1) and (1,1) and even powers (-1,1) and (1,1).

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All equations in this tutorial were created with QuickLatex

All graphs are created with www.desmos.com/calculator

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Here's a list of posts created so far on the topic of Functions:

1. Drawing piecewise defined functions
2. What are Polynomial Functions?
3. Power Functions: Case 1: a = n (n = 1,2,3...)

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