Welcome to a lesson on the intercepts and turning points of polynomial functions.
Sometimes these topics are referred to as short-run behavior of polynomial functions.
We are given a polynomial function of degree n. This gives us information about the graph
of the polynomial function. The function has at most n X intercepts, or horizontal intercepts,
which means it also has at most n real zeros. The function has at most n minus one turns.
Let's take a look at some examples. Here we have a degree four polynomial function,
which means the graph will have at most four X intercepts. In this case,
notice how the graph does have exactly one, two, three, and four X intercepts. And again,
because the degree is four, the function will have at most n minus one, or in this case, three turns.
Again, looking at the graph of our function, notice how the function turns here, it turns here,
and it turns here. So we do have exactly n minus one turns for this polynomial function.
Notice that when the graph turns, we either have a high point or a low point on the graph,
which will always give us a maximum or minimum function value at that location.
Let's take a look at another degree four polynomial function. Again,
the degree is four, but notice in this case that even though it is a degree four function, we
only have one and two X intercepts. This is still acceptable. Remember, it says at most
four X intercepts, so having two intercepts still satisfies the conditions listed above.
Also, notice this degree four polynomial function only has one turn here. So it
doesn't have exactly n minus one turns; it has at most n minus one turns. Therefore,
anything less than three turns and the polynomial function could still be of degree four.
One last thing to mention here about the end behavior: notice how the leading coefficient
in this first example is positive one. Because we have an even degree and a
positive leading coefficient, as X moves to the right, or approaches positive infinity,
or as X moves to the left and approaches negative infinity, notice how in both cases
the graph is moving up. Therefore, in both cases, F of X approaches positive infinity.
However, in the second case, when the degree is even and the leading coefficient is negative,
as X approaches the right or approaches positive
infinity, and as X approaches the left or approaches negative infinity, notice in both
cases the graph is going down. Therefore, F of X is approaching negative infinity in both cases.
Now, let's take a look at two examples of a polynomial function that has an odd
degree. Here we have a degree five polynomial function, which means we can have at most five
X intercepts. Let's see how many we have. We have one, two, three, four, and five, so we
have exactly five X intercepts, which is the most we can have for a degree five polynomial function.
Now let's see how many turns we have. We have a turn here, one, two, three, and four. Remember,
we can have at most n minus one turns. Five minus one is equal to four. So in this case,
we have the most possible turns for a degree five polynomial function.
For a second example of a degree five polynomial function, again we can have at most five X
intercepts and four turns. Notice in this case we only have one, two, and three X intercepts
and one, two turns. So if we didn't have the equation for this polynomial function, we
might think this function would only be a degree three polynomial function, which is possible. A
degree three polynomial function could have the same short-run behavior as this function here.
However, because we have the equation, we know it is a degree five polynomial function.
Let's finish by talking about the end behavior of these odd degree polynomial functions. Notice
how for this first example, the leading coefficient is negative one. Whenever
we have an odd degree polynomial function and the leading coefficient is negative,
as X moves to the right or as X approaches positive infinity,
notice how the function is going down. F of X is always going to approach negative infinity.
As X approaches the left or approaches negative infinity, notice how the graph
goes up. So as X approaches negative infinity, F of X will always approach positive infinity.
It's just the opposite when the degree is odd and the leading coefficient
is positive. As X approaches positive infinity, notice how the graph goes up,
so F of X approaches positive infinity. As X approaches negative infinity, or as X moves left,
notice how the function goes down. So F of X approaches negative infinity.
We had a whole other lesson on end behavior, or long-run behavior,
but I wanted to include this as a review.
Okay, I hope you found this lesson helpful.