Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

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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.