 So in this lecture we've been talking a lot about the behavior of polynomials we talked about the in behavior What happens is go to the far right and the far left We talked about the factor theorem and how it's related to the x-intercepts of the function The next thing I want to talk about is the idea of a turning point That is places on the graph where it changes directions now We've talked about these before turning points are really just gonna be our local Minima and our local maxima or the local extrema that we've talked about before So you see this idea about turning points be aware. This is nothing different than what we've talked about before We're talking about our local extrema the thing that I want to make clear about turning points though is that in this situation a Turning point on a polynomial is always going to be a smooth transition. You don't have something like like Nothing like lightning bolts or zigzags or whatever when you turn directions on a polynomial It's always going to be a smooth transition Always a rounded turn. Oh, it's nice and smooth It doesn't poke you when you try to patch your polynomial right here The number of turning points the number of extrema on a polynomial graph if f is a polynomial degree in the Number of turning points will be at most In I let me let me say this again the number of x intercepts on F is at most in Okay So a function can never have more than in x intercepts on its graph a Consequence of this has to do with turning points a consequence that every polynomial has a most x or has at most in X intercepts where in is its degree if the number of turning points is at most in minus one You can't turn directions more than in minus one times and the idea behind it is essentially the following I mean admittedly that the true argument comes from calculus here I mean we look at the derivative of a polynomial, which is itself a polynomial But if we had like say for x intercepts, what are the possibilities? We can kind of get something like this In order to match up with for x intercepts best case scenario in terms of turning We kind of get like four turns in there are three turns in the graph there one less than the degree there And again that the argument is a little bit more complicated than that But when you're graphing a polynomial, you cannot put too many turns on the graph You can't have more turns then one less than the degree of the polynomial And so with that in mind, we are now ready to play our everyone's favorite game show. Who's my graph? Imagine we have a polynomial function listed here on the screen We have f of x equals x to the fourth plus five x cube plus five x squared minus five x minus six Now our eligible Bachelorette today she's looking for a graph to complement her formula now Let's learn a little bit about our bachelorette today She is a polynomial of degree four and her leading term is x to the fourth on that's so great So from this we can learn a lot about her in behavior her Constant term negative six right here I want to mention corresponds to her y intercept because if you have a Constant pollen or if you have a polynomial you plug in zero to find its x intercept you plug in zeros for all of the x's But as the constant term doesn't have a zero, that's the only thing that resides That's only thing left over when you plug in zero to find the x intercept And so now let's meet our eligible bachelors today We have four graphs that want to be matched up with f of x right here So what can we say about these functions? What are some things we can say well? Are so these are some of the questions that are our functions going to ask her graphs right here So graph graph a what are you compatible? Are you a good match for our function f of f of x right here? Well? A here is going to say things like the following well when you look at my in behavior I have the in behavior of an even function. I point up on the left-hand side I point up on the right-hand side. So I'm an even degree function with a positive leading coefficient Oh, our our function f is swimming over such a comment right there Function a also tells us I should say graph a also tells us that I have exactly three turning points on my graph Which indicates that my degree does not exceed four. Oh, oh our lady here is really loving this so far This is so great. This is so amazing So good and so so far she should put little notes right here. She really likes she really likes graph a He's got the right leading term. That's exactly what what she needs It points up on both right and left-hand side has the right number of turning points. It's it's no more than three That's great Well when we come over and look at graph B graph B actually proposes to be the exact same way He's like, uh, I'm graph B right here. I also am a positive Even-degree polynomial. So on my right-hand side, I point up on my left-hand side. I also point up I am a true optimist and on that degree. I'm always pointing up my number of turning points is also three Which is one less than four, which means my degree does not get bigger than four. Wow Such good options for our function f today. She's looking both Graph a in graph B when we come over to graph C, though, let's see. I mean he Tries to tries to act like things are going on right here. It's like, oh, I'm a I'm a positive even degree polynomial as well I'm pointing up on the left. I'm pointing up on the right That means my leading coefficient is positive and my degree is even but and so yeah our function f She really likes that but then when she counts the number of turning points that he has one two three four five If you have five turning points, that means the degree in is at least six That's not compatible with our with our bachelorette right here. So she's gonna turn down graph C He is not the right choice for her Too many turning points too many turning points. She only wants a graph who is gonna have at most three turning points We look at graph D and the first thing she sees is that graph D does in fact have three turning points Just like she would just like she would like but wait a second Well, the graph is point the end behavior is pointing in the same direction That does indicate to her that this function is an even degree polynomial turns out There's no odd polynomials whatsoever in our group today. That's great But notice everything's pointing down this would tell us that the leading coefficient of this graph here is Negative and so we got graph D here. It's like I hate everything. I'm a pessimist his leading coefficients negative And as such that's not gonna be compatible for For our lady right here. So she's gonna turn down graph D as well So she's got to come back to graphs a and b which they both had positive in coefficients. They both had even degree They wrote even degree with the turning points at most three which would indicate that yeah These could these are likely x the leading term is likely some positive coefficient times x to the fourth But how does she decide which one it is which one to choose here? Well, it turns out the finding the final description We're gonna use here is going to be that constant term right the constant term corresponds to the y-intercept When you look at graph a graphs graph a has a positive Y-intercept because it intersects the y-axis above the x-axis graph B under the hand We can see has a negative x or y-intercept So if everything created equal B turns out to be the most compatible function for F right here It had the correct in behavior pointing up on the right it can point it up on the left It had the correct number of turning points It had it most three now just to be clear not every x to the fourth graph will have three turning points It could have only maybe one turning point But you can't have more than three turning points if your degree is four and then finally it had the correct y-intercept And so therefore It our our winner of today's game who is my graph is gonna be graph B And so our function the graph are gonna go on a very romantic date to the to the most romantic Restaurant in town and we'll let them we'll let them go have their time together And that also brings us to the end of our lecture today lecture 23 in our series about polynomial functions everything we talked about in this lecture was to help us better understand Graphs of polynomial functions as illustrated in our little game show we played here in the next lecture We're gonna we're gonna practice We'll we'll learn a little bit more about things that go into graphing polynomials And then we'll put all of that to the test and start grabbing graphing our own polynomials So stay tuned for our next episode which hopefully you should see the link for that on the screen right now