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Integer Power Functions - Concept
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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
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.
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
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.
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.
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.
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