If the given term is the dominating term of a polynomial function...

Question

If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a)domain (b)range (c)end behavior (d)number of zeros (e)number of turning points 10x7

Accepted Answer

Hey, everyone in this problem, a certain polynomial function has the following leading term 20 X to the exponent nine based on the leading term. And for the following features of its graph, one, the domain to the range three, the end behavior four, the number of zeros and five, the number of turning points, we're given four answer choices. Option A one negative infinity to infinity to negative infinity to infinity. Three, F of X goes to negative infinity as X goes to negative infinity and F of X goes to infinity as X goes to infinity, four up to 95, up to eight. Option B one negative infinity to infinity to negative infinity to infinity. Three F of X goes to infinity as X goes to negative infinity and F of X goes to negative infinity. As X goes to infinity, four, up to nine and five up to eight. Option C one negative infinity to zero, 20 to infinity. Three, F of X goes to negative infinity as X goes to negative infinity and F of X goes to infinity as X goes to infinity four up to nine and five up to nine and finally, option D one, negative infinity to 020 to infinity. Three, F FX goes to negative infinity as X goes to negative infinity and F of X goes to infinity as X goes to infinity or up to nine and five, up to 20. Now we have our leading term 20 X to the exponent nine. There's a couple of things we wanna notice about this leading term. And the first is the degree of this polynomial. So the degree of our polynomial is going to be the exponent on the leading term and the highest exponent in this case, it's not. So we have a degree nine and this is an odd number. So we have an odd degree polynomial and we have a positive leading term. OK? The coefficient is 20 that's a positive number. And so we have a positive leading term. All right. So the question is asking for the domain, the range and the end behavior the first three things. And I'm actually gonna start with the end behavior because I think that that helps us to determine the domain and range. So we have an odd degree polynomial with a positive leading term. I'm gonna draw a little sketch, I'm gonna draw the X and Y axes and we're gonna figure out where this thing is coming from and where it's going. OK. Well, we have a positive leading term with an on degree polynomial. We're gonna have the polynomial going down to the left in quadrant three and up to the right and quadrant one. Hm. And you can think of this if you forget what happens with an on degree polynomial with a positive leading term. Try to think of a function that you know the graph for think of say X cubed X cube might be a simpler one that you know the graph of and you know the general shape. It's gonna start down in quadrant three. It's gonna go up into quadrant one when you have a positive bleeding term. OK. So now we have our end behavior here and we can also figure out our domain and range. So let's write the domain and range first. So part one is the domain. Now the domain, we can have any X value, OK? We have a polynomial or any polynomial, we can have any X value. And so the domain is going to be negative infinity to positive infinity for the range. And we can see from the little diagram we drew that we have values going off to negative infinity from quadrant three and up to positive infinity from quadrant one. OK. So these values are going to either end of the spectrum. And so our range is also gonna be negative infinity to positive infinity. And this is true for any odd degree polynomial. Now, for part three, the end behavior we can use our little sketch to write that out. So we see that as X goes to negative infinity, our function F FX is also going to negative infinity. And similarly, as X goes to positive infinity, our function F of X is going to positive infinity. And again, these rules are true for any odd degree polynomial with a positive leading term. OK? Just like a line just like execute. Now, let's keep going. We've done the first three parts and now we're asked to find the number of zeros. Now, we're only given the leading term 20 X to the exponent nine. So we can't find the exact number of zeros, but we can find a bound on the zeros. OK? We have a degree nine polynomial which tells us that the maximum number of zeros we can have is nine. Hey, the maximum number of zeros is always the same as the degree. We have a maximum number, the zeros of nine. OK? And again, if you ever forget, think about the functions, you know, if we have Y equals X align, we know that that has 10, we have Y equals X squared, we know that that can have potentially two zeros. OK? So think about those functions, you know, all right, one more step five. And this is gonna be the max number of turning points. OK? Well, it's asking for the number of turning points, but again, just like the zeros, the leading term is not enough to determine exactly the number of turning points. But we can put a bound on them. Now, in this case, if we have a degree N polynomial, the maximum number of turning points is going to be N minus one. OK. So here we have a degree nine polynomial which tells us that the maximum number of turning points is going to be eight. All right. Sometimes people can get confused whether it's the same as the degree or one less than the degree. you can always think of your graph, you know your end point. So you could kind of draw out what happens if this thing has nine turning points. And if it had nine turning points and it started kind of in quadrant three, then it would end up in quadrant four, it would end, the end behavior would be different, it would be going down in quadrant four. OK? So if you use your end behavior, you can get a sense of how many turning points it should have as well or at least whether it should be odd or even. All right. So we were given just the leading term of this polynomial and we were able to determine all of this information, which is really great. Let's look at our answer traces. So we found that the domain of this function is from negative infinity to positive infinity. The range is also from negative infinity to positive infinity. We found that F FX goes to negative infinity as X goes to negative infinity. And F FX goes to positive infinity. As X goes to positive infinity, we found that we can have up to nine zeros and up to eight turning points. And so the correct answer choice here is option A. Thanks everyone for watching. I hope this video helped see you in the next one.

If the given term is the dominating term of a polynomial function, what can we conclude about each of the following features of the graph of the function? (a)domain (b)range (c)end behavior (d)number of zeros (e)number of turning points 10x7

Key Concepts

Dominating Term in Polynomials

End Behavior of Polynomials

Zeros and Turning Points

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