Find a rational function ƒ having a graph with the given features...

Question

Find a rational function ƒ having a graph with the given features. 
x-intercepts: (-1, 0) and (3, 0)
y-intercept: (0, -3)
vertical asymptote: x=1
horizontal asymptote: y=1

Accepted Answer

Hey, everyone in this problem, the characteristics of the graph of a rational function F are given below. And we are asked to find that function F of X. OK. So we're given the X intercepts Y intercept, one of the vertical asymptotes and a horizontal Asymptote. We're gonna come back to all of those conditions and work through them one by one. As we sort out this problem, we're given four answer choices A through D, all four of them have the equation of a rational function in factored form A and they just differ in what those factors are and what the coefficients are. So let's start with the first piece of information we're getting. OK. We're told that we have X intercepts a negative two zero and five comma zero. Now, in a rational function, the only way to have X intercepts is if the numerator shares those X intercepts, OK. If we imagine setting our function F of X equal to zero, that can only happen if the numerator is equal to zero. OK. So this tells us that the numerator must have the same factors are these same X intercepts. We'll say negative two comma zero and five comma zero. Now, if we have a function that has these X intercepts, that tells us something about the factors of the function. OK. If the numerator has these X intercepts and we write the numerator in factored form, then X plus two and X minus five must be factors of that numerator. Why is that? Well, if you imagine again, setting that function equal to zero, OK, you're gonna factor it to find the X intercepts. If we have the factor X plus two, we set that equal to zero, then we're gonna get X is equal to negative two, that first root or that first X intercept. OK. If we set X minus five equals to zero, we get X is equal to five, that second X intercept. OK. So what we're doing here is kind of working backwards. Normally we have a function, we want to factor it to find the X intercepts here. We're given the X intercepts and we're working back backwards to write down those factors. OK. So our X intercepts have told us some information about the numerator. Let's move on. And we're actually gonna skip the Y intercept for now. We're gonna come back to that. And let's look at these vertical asymptotes. So we're giving one vertical ascent. OK. So we have a vertical ascent to at X equals two. OK? And recall that a vertical Asymptote is going to occur where the denominator of our simplified function is equal to zero. So this is gonna be really similar to the X intercept where we worked in the numerator to find factors except now we're working in the denominator to find factors. OK? When the denominator is equal to zero, X is equal to two or vice versa. When X is equal to two, the denominator is equal to zero. OK? What this tells us is that the denominator ha the route of X equals two. OK? Because when X equals two, the denominator is zero. And just like we did for the numerator that tells us that X minus two is a factor. And again, we're talking specifically about the denominator because we're looking at that vertical asym to, all right. So we have some factors in the numerator from our X intercept. We have a factor in the denominator for from our vertical Asymptote. Let's move to our horizontal ascent tote. And we're told that that is that Y is equal to two. So we have a horizontal asom tote Y is equal to two. Now, if we have a horizontal ascent that is not Y equals zero, that tells us that the degree of the numerator of our function must be equal to the degree of the denominator of our function. OK? So we know that the degrees have to be equal. We also know that if we have this case, then the horizontal Asymptote occurs at the ratio of the leading coefficients. So the leading coefficient of the numerator divided by the leading coefficient of the denominator must be equal to two because that's where our horizontal assum two is. OK. So this gives us some information about the relationship. Now between the numerator and denominator, the other information we had was strictly about one or the other. This gives us some information about the relationship between those two. So let's write out what we have so far. OK? I know we haven't touched that Y intercept yet, but we're gonna write our function so far and then apply our, our Y intercept or see if our function matches that Y intercept. So what we have is a function F of X, it's gonna have some leading coefficient. We're just gonna call K from the X intercept. We determine that it has two factors in the numerator X plus two and X minus five. In the denominator, we found that it had a factor X minus two. Yeah. Now that's the only factor we found in the denominator. But from our horizontal axon to, we know that the degree of the numerator has to be equal to the degree of the denominator. In the numerator, we have two linear terms multiplied together. So this is a degree two polynomial in the numerator, which means we need a degree two polynomial in the denominator. So we actually need to multiply that denominator by another factor. And for now we're just gonna call it X plus C because we don't actually know what that factor should be or what it is. Now from our horizontal ascent to we also know the ratio of leaving coefficients must be equal to two. OK. In this case, the leading coefficient of the numerator in our function is K in the denominator, it's just one. OK? We're gonna have X multiplied by X that gives us X squared for our leading term. That's just a coefficient of one. So our leading coefficient, the ratio sorry of the leading coefficients is gonna be K, we know we want that equal to two. And so our function is gonna be two multiplied by X plus two, multiplied by X minus five, divided by X minus two, multiplied by X plus C. All right. Now we're gonna use that Y intercept information to find this value of C. OK? We know that we have a Y intercept or we're told that we have a Y intercept at zero comma negative seven. OK? We're gonna substitute that into our equation. And so for sea, so we got F of X which is negative seven is equal to two. When X is equal to zero, the numerator is gonna be two multiplied by two multiplied by negative five. The denominator is gonna be negative two, multiplied by six. We're gonna multiply both sides by negative to C. OK? So we got 14 C is equal to two multiplied by two multiplied by five which is negative 20. This tells us that C is equal to negative 20 divided by 14, we can divide both of those by two to simplify. So if we divide the numerator and the denominator by two, we get that C is equal to negative 10 divided by seven. So now we have used all of our information, we got information about the roots of the numerator from the X intercept. We got information about the roots of the denominator from the vertical Asymptote. We had information about the leading coefficient from our horizontal Asymptote as well as the degree of the numerator and denominator. And finally, we used our Y intercept to figure out that extra term in the denominator. And we found that our rational function F of X is equal to two multiplied by X plus two, multiplied by X minus five, divided by X minus two, multiplied by X minus 17th. And that is our final answer for this problem. Now, if we go back up to our answer choices and we compare what we've just found, we can see that this corresponds with answer choice. A thanks everyone for watching. I hope this video helped see you in the next one.

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Find a rational function ƒ having a graph with the given features. x-intercepts: (-1, 0) and (3, 0) y-intercept: (0, -3) vertical asymptote: x=1 horizontal asymptote: y=1