finding function with given asymptotes, etc

[sunaga]

How do I do the following problem? Any help is greatly appreciated.

Write an equation for a function whose graph has a horizontal asymptote of y=-2 and vertical asymptotes of x=1 and x=0. The function must have at least one relative extremum and at least one point of inflection.

Thank you very much in advance.
Sundar Naga

 

[sunaga]

How do I do the following problem? Any help is greatly appreciated.

Write an equation for a function whose graph has a horizontal asymptote of y=-2 and vertical asymptotes of x=1 and x=0. The function must have at least one relative extremum and at least one point of inflection.

Thank you very much in advance.
Sundar Naga

By making sure that it has the desired asymptotes I get the following for the required function:
-2x^2/(x (x-1))
Now, how do I make sure that it has at least one relative maximum or minimum and at least one inflection point?

Thank you very much in advance.
Sundar Naga

 

[Steven G]

First make a graph of a function that has the required conditions. Then you can try to figure out the function.

 

[Steven G]

y= -2x^2/(x (x-1))
Now, how do I make sure that it has at least one relative maximum or minimum and at least one inflection point?

How do you normally find a relative min or max? Do that same procedure and if you find one (or more) relative extrema, then you are ok.
How do you normally find a point of inflection? Do that same procedure and if you find one (or more) POI, then you are ok.

 

[topsquark]

By making sure that it has the desired asymptotes I get the following for the required function:
-2x^2/(x (x-1))
Now, how do I make sure that it has at least one relative maximum or minimum and at least one inflection point?

Thank you very much in advance.
Sundar Naga

This does not have a vertical asymptote at x = 0.

My suggestion is to look at the asymptotes separately. For example, start with y(x) = A/x + B/(x - 1) and work up from there.

-Dan

 

[Steven G]

Personally I would consider using a piecewise function basically putting the pieces together. Possibly in the end you might see a non-piecewise function

 

[sunaga]

Thank you Dan. So, if there are 2 or more vertical asymptotes, I should consider as many terms separately. Correct?

Sundar Naga

 

[sunaga]

Yes, Dan. I tried 3 terms with vertical asymptotes at 0, 1 and 3. I really like how the curve shows these asymptotes. Thanks a lot.

Sundar Naga

 

[JeffM]

Personally, i like Dan’s general approach of building from the ground up. Moreover, I suspect the objective is to create a function that is not defined piecewise.

What kind of functions have vertical and horizontal asymptotes?

Rational functions may.

[math]r(x) = \dfrac{p(x)}{q(x)}, \text { where } p(x) \text { and } q(x) \text { are polynomials.}[/math]
And q(x) must have x and (x - 1) as factors. If we want the function to have no other vertical asymptotes, q(x) must have no other factors. So

[math]r(x) = \dfrac{p(x)}{x^m * (x - 1)^n}, \text { where } m, \ n \in \mathbb Z^+.[/math]
Question, if a rational function has a horizontal asymptote other than 0, what do we know about the degree of p(x)?

 

[Steven G]

No no no, don't assume anything, especially if it will make your life easier. It never said you can't use a piecewise function.