Integer Power Function

Power functions are functions where y = x^n where "n" is any real constant number. When "n" is a positive integer, we have two possible scenarios of an integer power function. When "n" is odd, the function passes through the origin, (1,1) and (-1,-1). Also, as the exponent increases, the function becomes steeper. When "n" is even, the function passes through the origin, (1,1) and (-1,1). These functions are symmetric about the origin.

I want to talk about power functions. I specifically want to talk about power functions where the exponent is a positive integer. We can divide these into two cases.

First, the odd power functions, which are Y equals X to the N where N is an odd number. 1, 3, 5, et cetera, and there are the even power functions where Y equals X to the N and N is an even number, 2, 4, 6, et cetera, let's look at Geometer Sketch Pad to get an idea for what these functions look like.

Okay. Here we're in Geometer Sketch Pad. We're looking at odd power functions right now. You can see I've got graphed Y equals X, Y equals X cubed Y equals X to the fifth and Y equals X to the seventh. They're color-coded so you can tell which is which and I can change the power of this guy so we can look at other examples of power functions X to the 9th and X to the 11th, et cetera.

Notice that they all seem to have things in common. One of them is that they all pass through the origin. They all pass through the point 1,1 All pass through the point negative 1, Negative 1. They all have an increasing tendency. They go up from left to right.

You'll also notice that as the power goes up, as we go from X to X cubed to X to the 5th, X to the 11th and so on, that the graph gets closer and closer to the X axis between 0 and 1. It's kind of sucked down into the X axis.

But as we zoom out, you can see that there's the opposite behavior. The higher the exponent is, the faster the function increases. So this is X to the 11th right here. This is X to the 5th. X cubed. And X. And you can see that X to the 11th increases really fast. And if I increase to an even higher exponent you can see it's faster still.

Now let's look at even power functions. Here the even power function is X squared, X to the 4th, X to the 6th and X to the 8th. Like the odd power functions they all pass through the origin. They all pass through 1, 1, but these graphs also pass through negative 1, 1, and they are not increasing functions. In fact, they're symmetric about the Y axis.

Now, like the odd power functions, the higher the power is, the more the graph hugs the X axis between 0 and 1. You can see that's happening here with Y equals X to the 8th. If I increase the power, that's even more noticeable, X to the 10th. X to the 11th, X to the 12th and so on. Let's see what happens when we zoom out.

Again, like the odd power functions, the higher the power, the faster the increase when X is greater than 1. So this is X to the 12th. This is X to the 6th. X to the 4th and X squared.

So just a brief overview. You can see that this graph is symmetric with respect to the Y axis. What kind of symmetry do the odd functions have? Let's take a look back at them.

These guys are symmetric about the origin. We say they have 180 degrees of symmetry with respect to the origins. You can rotate them 180 degrees and you'll get the same graph. Another way you can look at it is you can reflect them across the X axis and then across the Y axis, and you'll get the exact same graph. Okay.

Let's review what we've learned. So about odd functions. We've learned that, first of all, let's talk about the domain. You can see that these functions are going to be defined for all real numbers. There's no special reason why they wouldn't be. So the domain is going to be all real numbers. From negative infinity to infinity.

Also the graphs pass through 0, 0, and 1, 1. Graphs include 0, 0 and 1, 1. And for the moment I'm not going to list negative 1, negative 1 because so far what I've written also goes for even power functions. The domain is all real numbers for these guys. And they also pass through 0, 0 and 1, 1.

So I'm going to focus on things on properties that they both have. And speaking of properties that they both have, if you look at the right side of either kind of power function, the right side, the end goes up, the end behavior is as X goes to infinity Y goes to infinity. Let me write that down. The right end goes up. Whoops. One letter ahead. End goes up.

Now, properties that are specific to the odd and even ones, the odd power functions are symmetric with respect to the origin. Let me write that down. Symmetric with respect to -- whoops -- respect to the origin. Okay.

Remember, that kind of symmetry is where you can rotate the graph 180 degrees and you'd get the same picture. Also, the range the all real numbers. You can see it looking at the graph that these are going to go up to infinity and down to negative infinity. So you're going to hit every possible Y value here.

Now, let's take a look at the even functions. Here we have symmetry with respect to the Y axis. So symmetric with respect to, I'll abbreviate, the Y axis. And we also have -- we don't have the same range here. We can't get negative numbers out of this function. When you're raising to an even power, negatives become positive. So the range only includes the non-negative numbers. you can get 0. And you can get positive numbers but you can't get negatives.

So just to recap, all of these power functions, I'm speaking specifically about power functions where the exponent is a positive integer. All of them have a domain, all real numbers. All of them pass through the point 0, 0 and 1, 1. All of them have the property that the right end goes up.

Now, if you want to know what the left end does you look at the symmetry, when a graph is symmetric with respect to the origin, the left end goes down. The range is all real numbers. With even functions, they're symmetric with respect to the Y axis. So the left end does the same thing the right end does. It goes up to infinity and the range is the set of non-negative numbers.

We'll be needing to understand power functions really well when we start setting polynomial functions, because it turns out that the end behavior of polynomial functions is determined by the end behavior of power functions.