Solve each problem. Work each of the following. Find an equation ...

Question

Solve each problem. Work each of the following. Find an equation for a possible corresponding rational function.

Accepted Answer

Hey, everyone in this problem, the characteristics of the graph of a rational function F are given below. We're asked to find E S we're told that there is an X intercept at zero comma zero that there are vertical asymptotes, X equals negative two and X equals positive two. And a horizontal Asymptote at Y equals zero. Option A F of X is equal to one divided by X minus two, multiplied by X plus two. Option B F of X is equal to X divided by X minus two, multiplied by X plus two. Option C F of X is equal to X divided by X plus two. And option D F of X is equal to X divided by X minus two. Let's start with our X intercept. So we have an X intercept at zero comma zero. Recall we have this point like this that tells us that when X is equal to zero F of X is equal to zero. Now we can imagine we have a rational function if our function F of X is equal to zero, that can only occur if the numerator is equal to zero. OK. So the new writer must be equal to zero when X is equal to zero. OK. So the numerator must have a root of a syrup. OK. When I say root, you can call it a zero as well. If you want to, if it has a root of zero, it means it has to have a factor of X. OK. So the numerator must have a factor of X. OK. So it has a factor of X which means when you set it to zero, you get a zero and zero. True. All right, great. So we figured out what we need in order to get our X intercept that we want. Let's move to our vertical asy to. So recall that the vertical Asymptote is going to occur where the denominator is equal to zero. OK. So if we have X equals negative two and X equals positive two, then the denominator must have these values as roots by definition, these occur where the denominator is equal to zero. So it's like looking for the roots of the denominator. And so these must be roots. Now, if these are roots similarly to our explanation for X intercept, then we must have the factors X plus two and X minus two in the denominator, hm If X equals negative two is a root, then X plus two is a factor because when we have X plus two and we said it equal to zero, we get X is equal to negative two and similarly with X minus two, if this is a factor we set it equal to zero, we get X is equal to two, which is that second vertical as to. So we're gonna have these two factors in the denominator to account for each of those vertical asci totes, we have one more piece of the puzzle to look at and that is our horizontal asset to. So our horizontal as to we're told occurs at Y is equal to zero. Now, when we have a horizontal ascent tode at Y equals zero, we call, this means that the degree of the denominator of our rational function is greater than the degree of the new murder. OK. So we need that degree of the denominator to be bigger. So let's go ahead and write what we have so far and make sure that it matches with this horizontal as tone. So we have F of X, we know that we have to have a factor of X in the numerator. So let's write that and we know in the denominator we have to have factors X plus two and X minus two. If we write all that and we look the degree of the numerator is one and we just have an X term, the highest exponent is one. So our degree is one in the denominator, we have X plus two multiplied by X minus two. If we were to expand, we would get an X squared term. OK. So the highest exponent is two. So the degree is two. So this also satisfies that horizontal asy component. OK. The degree of the denominator is bigger than the degree of the numerator already without doing anything else. And so this function F FX satisfies all of those conditions. OK? It has an X intercept at zero, it has vertical asymptotes at X equals negative two and X equals two and it has a horizontal Asymptote at Y equals zero. And so the correct answer here is the function F of X is equal to X divided by X plus two, multiplied by X minus two, which corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

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Solve each problem. Work each of the following. Find an equation for a possible corresponding rational function.

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