Question

Use the graph below to construct the rational function that it represents. The thicker blue lines represent the graph and the thinner red lines represent the asymptotes. Please show all of your work. (see attached graph) Use the graph below to construct the rational function that it represents. The thicker blue lines represent the graph and the thinner red lines represent the asymptotes. Please show all of your work. (see attached graph) @Mathematics

Answer 1

I'd like to know myself how to do it

Answer 2

maybe we can find someone to help

Answer 3

\[\large f(x) = -\frac{x^3}{-x^4+81}\]

I never did this kind of exercises, so I found it a lil' bit challenging, but it's kind of funny.

Okay, observing the graph, we get that it has quite a few problems at -3 and 3. So my first guess was put x^4-9 at the denominator so that -3and3 aren't allowed.

Now, the central graph, is just a x^3 (elementary function) but reversed, so I put a minus sign in front of it.

nevertheless, we see y = 0 to be an horizontal asimptote, so the denominator had to be a degree higher than the numerator, so that when we do the limit for x to infinity we get 0.

So of course, I put a higher degree at the denominator, and I made it 81 so that the solution'd came out properly.

Answer 4

sorry, my first guess was to put x^2**-9. Correct myself up there :)

Answer 5

This looks like
\[y=\frac{x}{(x-3)(x+3)}\]

It has two poles (goes to infinity) at x=3 and x=-3. So expect (x-3)(x- -3)= (x-3)(x+3) in the denominator. It has one zero, so expect x in the numerator. But that is as much as I know about justifying this guess. However, plotting it shows it has the correct behavior.

Answer 6

so which one of you is correct? lol

Answer 7

both are fine I guess, they have the same behaviour and you can probably find few more that do that same thing.

Answer 8

Great! Thank you!