In this video, we will look at four examples
of determining the end behavior
or long run behavior of power functions.
For a quick review, as X approaches negative infinity,
we are moving left along the graph.
And as X approaches positive infinity,
we are moving right along the graph.
And if the function values, or Y values,
go up, the functions as your Y values
are approaching infinity.
If the function values, or Y values,
go down, the function values are approaching
negative infinity.
Notice for these first two examples,
the degrees of the power functions are even.
Our first function is F of X equals X to the sixth.
Notice as X approaches negative infinity,
or as they move to the left along the graph,
the graph goes up and therefore the Y values
or function values approach infinity.
Which means as X approaches negative infinity,
F of X approaches positive infinity.
And as X approaches positive infinity,
or as we move to the right along the graph,
notice how the graph goes up
and therefore the function values, or Y values
approach positive infinity.
As X approaches infinity, F of X approaches infinity.
And now to take a look at F of X equals negative
X to the fourth.
As X approaches negative infinity, or as we move
to the left along the graph, the graph goes down
and therefore the Y values or function values
approach negative infinity.
As X approaches negative infinity,
F of X approaches negative infinity.
As X approaches positive infinity,
or as we move to the right along the graph,
notice how the graph goes down
and therefore the Y values or function values
approach negative infinity.
As X approaches infinity, F of X approaches
negative infinity.
Again, notice how for both of these power functions,
the degrees are even.
But for the first function, F of X equals X to the sixth,
A is one or positive, and for the second function,
F of X equals negative X to the fourth,
A is negative one, or negative.
Whenever we have a power function with an even degree
and A is positive, this will always be the end behavior
and when the degree is even and A is negative,
this will always be the end behavior.
Now let's take a look at two examples
where the degree is odd.
Our first function is F of X equals X cubed.
As X approaches negative infinity,
or as we move to the left along the graph,
notice how the graph goes down,
and therefore the Y values, or function values
approach negative infinity.
As x approaches negative infinity,
F of X approaches negative infinity.
As X approaches positive infinity,
or as we move to the right along the graph,
the graph goes up,
and therefore the Y value or function values
approach infinity.
As X approaches infinity, F of X approaches infinity.
And now let's take a look at F of X equals
negative X to the fifth.
As X approaches negative infinity,
or as we move to the left, notice how the graph goes up
and therefore the Y values or function values
approach infinity, as X approaches negative infinity,
F of X approaches positive infinity.
As X approaches positive infinity,
or as we move to the right along the graph,
the graph goes down and therefore the function values,
or Y values approach negative infinity.
As X approaches infinity, F of X approaches
negative infinity.
So again, notice how both of these power functions
have an odd degree, but for F of X equals X cubed,
A is one or positive, and for F of X equals
negative X to the fifth, A is negative one, or negative.
Whenever we have a power function with an odd degree,
and A is positive, this will always be
the end behavior, and when the degree is odd
and A is negative, this will always be the end behavior.
Let's summarize this with a table.
When we have a power function in the form of
A times X raised to the power of N,
if A, the coefficient is positive,
we look at this row.
If A is negative, we look at this row.
And then if the degree is even, we look at this column.
If the degree is odd, we look at this column.
In our four examples, we looked at each
of these four cases.
I hope you found this helpful.