Hey, everyone. We've been looking at the different elements of the graphs of polynomial functions, like the end behavior as x approaches negative infinity or positive infinity and the zeros of our function and what our graph is doing at each of those zeros, whether it is crossing our x-axis, like at this point or simply touching and bouncing off like at this point. We have left to consider what's going in between all of these points as our graph is going from increasing to decreasing or decreasing to increasing and how many times it can do that. So, I'm going to show you exactly how to determine how many times our graph can change direction using one simple thing, the degree of our polynomial. So let's go ahead and jump right in.

These points where our graph is changing direction are called turning points, where a graph is going from increasing to decreasing or decreasing to increasing, really just from going up to down to down and back up. The maximum number of turning points that you can have is simply n−1, where n is the degree of our polynomial.

Looking at our first example down here, I have 6x4+2x. The degree of this polynomial is 4. If I simply take 4 and subtract 1, that gives me a maximum number of turning points of 3. We say maximum number because it doesn't have to have 3 turning points, but it just can have up to 3 turning points.

One other thing that we want to consider with our turning points is that they will either be a local maximum or a local minimum depending on what direction it is changing from. This point right here has lower points on either side. It is at the top of a hill. It represents a maximum. Whereas this point down here, it is going up on either side. It's in a little valley, so it is a minimum point. Now that's just something to consider about turning points, but let's go ahead and calculate some more maximum numbers of a couple more polynomial functions.

Looking at example b here, I have f(x)=x2-1. So the degree of this is simply 2. 2−1 gives me one maximum turning point, which makes sense because I know that the shape of this graph is like this, only one turning point, because it is a quadratic function.

Let's look at one final example here. So I have −x2+5x3-6x. Now remember, the degree is our highest power and this is not in standard form, so it's not my first term, but I have this 3 right here. That is my degree. 3−1 gives me a maximum number of turning points of 2, and that's all.

So you might be wondering what we're going to use turning points for, why would I need to know the maximum number of turning points. It's really just going to serve as a way to check that we have graphed a polynomial function correctly. So that's all you need to know about turning points. Let's go ahead and get to graphing.