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Answer cach of the following Suppose f(x)= N(r)Dlx) , where N(r) and Dlx) are polynomials and Dx) is not the zero polynomial. Under what condition(s) will the graph...

Question

Answer cach of the following Suppose f(x)= N(r)Dlx) , where N(r) and Dlx) are polynomials and Dx) is not the zero polynomial. Under what condition(s) will the graph of havc the x-axis as horizontal asymptote?

Answer cach of the following Suppose f(x)= N(r)Dlx) , where N(r) and Dlx) are polynomials and Dx) is not the zero polynomial. Under what condition(s) will the graph of havc the x-axis as horizontal asymptote?



Answers

Let $f$ be the function whose graph is obtained by translating the graph of $y=\frac{1}{x}$ to the right 3 units and up 2 units. (a) Write an equation for $f(x)$ as a quotient of two polynomials. (b) Determine the zero(s) of $f$ (c) Identify the asymptotes of the graph of $f(x)$

Okay, so we're looking for a function that has a vertical ascent owed at X equals to a vertical ascent up means that in the denominator There has to be an X -2 -2 because if you solve x minus two equals zero, then you get X equals two. So that's going to give you a vertical ascent to also needs to have a horizontal assam. Toad at Y equals negative two. So if I put a negative two X up here, then the degree of the numerator equals the degree of the denominator. And so the horizontal assume tote is just at the leading coefficient of the numerator divided by the leading coefficient of the denominator, Which is -2. And also f of zero equals 0. Um Well let's see F of zero is negative two times 0/0 minus two. Um That does indeed give us zero. So there it is.

Okay. So um let's find uh horizontal a stem toad. Mhm. So mm. We know that an equals to two And m equals to two. Okay. From what we learned from the textbook, we know that if any goes to m mhm. Yeah. The lie 3/1. Or you can you have in the last three? Why you go to the three? Is the horizontal assembly thought. Yeah. Of mm hmm, affects Okay, So for part B Let's let's find the .1. H. F X crosses horizontal. A sum total. Okay, so Let H. FX equals 2 3. Okay, so we have three x squared plus. Yeah. Yeah. So we have okay, we multiply X uh square plus three to the both sides of the equation. Okay, so we have three X squared plus nine. Sure. So we have N. x. equals two 14. So actually goes to 14/8, Which it goes to 7/4. Okay. So the graph, so I think we should also plug this back into our our equation, our function here. So H off 7/4. It goes to. Okay. Okay. So this goes to This is a coastal three. Okay. Yeah. Telegraph across this the horizontal a sample tote at 7/4. 3

Really given problem, we want to find conditions on A. B, C and D. So in this case we have F of X equals they act Crosby divided by C. Express B. Okay. In this case we want to make sure that there's no horizontal or vertical ascent oats. Well, the simple fix is just to let C40. And then it doesn't matter what anything else is because then you don't have a vertical ass from tow or horizontal ass from tow. However, if we want to do this a little bit more appropriately, what we could do is consider the fact that um there will be a horizontal as motto in this case of a oversea. So to avoid that, we need to make sure that this whole term is gone. Um So by letting C equals zero, we end up getting rid of this term. So that's basically how we explain this away. Unless C. C. Must Equal zero. In order for this to be the case. Now that we do this, we have our line and it doesn't matter what A. And B. And D do because he is zero. But the second we add this C. X. In and we choose the value of C cannot be zero. If it's one or negative one, we start running into issues with regards to horizontal and vertical ass and tits.

The question is giving us that D All right, listen good, great. Or is under for good his D X axis on 30 vertical assembled is the y axis. Now, from this information, we can conclude a few things. First, let us say that from the fact that the horizontal axis is the X axis, we can see that if the required rational function is the X bike UX isn't required. National function is the X like you x then? Because the horizontal axis is the excess is the X axis. So we can conclude that the degree Oh B X, the numerator of all union is less than the degree off que works, which is their dinner mutable. You know, mia Now, from the second fact that the vertical axis is the wire now the Y axis equation is X is equal to zero. So we know that Q X is equal to zero. Wheeler X is equal to zero. So we can simply say it. You can write two X is equal to one by it. Okay, No. Two X is equal just X. Because just that you nominated for you know, mia Now it can also people no means like X squared, X cubed and so on. But we need just one So we like for the simplest be exp i Q X that begin right now can simply be one upon X because both the conditions are not satisfied when we put X equals zero to denominator polynomial goes to you and also that the X, which is just one here is the degree off. P X is less. Thank you. Extra border conditions are met off course. We can write other answers pool which will satisfy the two conditions like we conveyed X plus one of one X square and so on. But the question as access wanted one answer so we can simply say that one more. Next is a rational function. It satisfies the conditions given


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