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, we're given characteristics of a graph of a rational function F and we're asked to find the function F of X. Yeah. So we're given X intercepts, we're given one of the vertical atom totes, we're given the horizontal ay tote. And we're gonna come back to all of those in the specifics. When we work through this problem, we're given four answer choices. ABC and D and all of them are rational functions. OK? That differ in the values they have in the numerator and the denominator. So let's start with our X intercepts. We're given the X intercepts six comma zero and eight comma zero. Yes. So we've been X intercept at six and X intercept at eight. Now, if we have X intercepts of our rational function F effects, that means that the numerator must have those X intercepts as well. Yeah. So the numerator must have these X intercepts. Yeah, because if you imagine setting F of X equals to zero to solve for the X intercepts, the only way we can satisfy that equation is if the numerator is equal to zero. OK. So for the numerator to be equal to zero, it must have those same X intercepts. So if we have a numerator with X intercepts six and eight, our numerator must have the factors X minus six and X minus eight. So we're kind of working backwards here. Normally, we have a function, we would factor it. And then to find the X intercepts, we would set that function equal to zero. OK. So if we did that, in this case, we set this function equal to zero, we have our two factors. OK? We would set each factor equal to zero. And so if we set X minus six equal to zero, we get that X is equal to six. And if we set X minus eight equal to zero, we get that X is equal to eight. So we have those two X intercepts that we wanted it. OK? So here we're just working the other way around. We were given the intercepts, we wanted to write down the factors. OK. So looking at our answer choices, we can already eliminate two of the options. We know we have to have X minus six and X minus eight. Those two factors in our numerator option A and option C have factors of X minus six and X plus eight, but they don't have that factor of X minus eight. OK. So we can already eliminate A and C. Now let's move to the vertical asymptotes. Now we're told one of the vertical asymptotes is that X equals five. Now recall that the vertical asymptotes occur where the denominator is equal to zero. If the denominator is equal to zero, that's equivalent to finding the roots. OK. So the denominator must have a root. OK? Or a zero at X equals five. Yeah, because when we set it equal to zero, we get X equals five, you must have a root or zero, X equals five. And similarly to with the numerator, if it has a root at X equals five, it must have a factor at X minus five or of X minus five. OK. So same thing we started with the zero because we know the vertical Asymptote occurs where the denominator is equal to zero. OK? So the denominator must have a zero or a root at X equals five, which tells us that X minus five is a factor. OK? So we need X minus five in the denominator of this function. We look at our answer choices, we can see that option D does not have this factor that we need, we can actually eliminate option D which is enough for us to choose option B as a correct answer. But I wanna keep going and look at the horizontal Asymptote. OK? Just so that you understand how to work this out. So if we look at our horizontal as to, we're told that this occurs at Y is equal to six. Now we have a horizontal Asymptote where Y is equal to some number that is not zero. What this means is that the degree of the numerator must be equal to the degree of the denominator of this function. OK. So using our X intercept, we found that we had to have at least two factors in the numerator. So we have at least a degree two polynomial. That means that in the denominator, we also need to have a degree too. So we needed an extra factor. Now, our horizontal Asymptote being at Y equals six also tells us that the leading coefficient of the numerator divided by the leading coefficient of the denominator must be equal to six. OK? If we have that the degree of the numerator and the degree of the denominator are equal, then the horizontal asom tote is given by this ratio of leading coefficients. OK? All right. So if we try to write out our function now with the information we're given, we know that in the numerator, we need X minus six and X minus eight and those two multiplied together in the denominator, we know we need a factor of X minus five and we need another factor of X. I'm just gonna say minus C. Yeah, we aren't given more information about that second vertical Asymptote. So we don't actually know what that value of C is gonna be. So we're just gonna write it as X minus C. And one more thing we know that the leading coefficient of the numerator divided by the leading coefficient of the denominator must be six. So we can achieve that by putting a six in the numerator, then we have six divided by one which gives us six. OK. So our function F of X must be in this form, we have six multiplied by X minus six, multiplied by X minus eight, divided by X minus five, multiplied by X minus C for some constant C. And if we compare this to our answer choices, we see that this corresponds with answer choice B F of X is equal to six, X minus six, multiplied by X minus eight, divided by X minus five, multiplied by X plus three. And so they've chosen that constant C of negative three here. And that equation satisfies all of the conditions we were given. Thanks everyone for watching. I hope this video helped see you in the next one.

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

Key Concepts

Rational Functions

Polynomial Functions

Asymptotes

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