In Exercises 10–13, use the Leading Coefficient Test to determine...

Question

In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).]  f(x) = x^6 -6x^4 + 9x^2

Accepted Answer

hey everyone in this problem we are given a polynomial function and we're asked to use the leading coefficient test to determine the end behavior of the graph fan to sketch the graph. Alright. So the function we're given is F of X equals to X to the six plus three X to the four minus five X squared plus one. Okay. Alright. So the first thing we want to do with our leading coefficient test and look at the degree of the polynomial in the sign of the leading coefficient. Okay, so the degree of the polynomial is going to be the highest exponent. Okay, now in this case the highest exponent is six. Okay, we have an X to the sixth term. That's our highest exponent. And so we get degree six. Okay, Now six is an even number. So we're gonna write that this is even then we have our leading coefficient and our leading coefficient is a coefficient that corresponds to the highest degree or the highest exponent term. The leading term in this case we have to Okay, which is positive, This is bigger than zero. So we have an even function. The positive leading coefficient. This tells us that as X goes to, let me write that as X goes to either positive or negative infinity. Our function F of X is going to go off to positive infinity. Okay, so both the left and right behavior at the ends is going to be up to positive infinity. Okay, so if we're looking at our potential answers here, we see that in a we have right and behavior going to positive infinity. That's what we have but we have left and behavior going to negative infinity. We don't have Okay. And B we have right and behavior going to positive infinity and left and behavior going to positive infinity. Okay, So that's a possibility in C we have the right and behavior going to negative infinity. So that's not a possibility. And indeed we also have the right and behavior going to negative infinity. So that's also not a possibility. Okay, so the only real possibility here given the end behavior is that we have answer B. Okay, let's go ahead and just see some more information about this to get an idea of the sketch without using the possible answers. Okay, let's look at this symmetry of this graph in this graph we have in our function we have only even terms. Okay, what does that mean? Only even terms means that the exponents on each of the terms we have is even So we have exponents 642 and zero. Okay, these are all even And what that tells us if we have only even terms is that we have y axis symmetry. Okay, Y axis symmetry means that the graph of our function is going to be symmetrical across the y axis. And you can see that on the left hand side of this graph, it's almost like a mirror across this Y axis to the right hand side. Okay, one other thing we could do to help us graph this is find the Y intercept. Okay? And in this case, why intercept one X zero in this case is going to be one. Okay. And you see the point here, 01 on this graph. So this is indeed the sketch of this function. Okay. And so to reiterate, we have answer B, R right and behavior as X goes to positive infinity, our function goes to positive infinity are left and behavior as X goes to negative infinity, our function goes to positive infinity. And our graph is the graph and B. We have y axis symmetry and Y intercept at one. Thanks everyone for watching. See you in the neck.

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In Exercises 10–13, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] f(x) = x^6 -6x^4 + 9x^2