rational function for this graph

[calcstudent123]

I have this graph here of a rational function.



It has x-intercepts at -3 and 1
and vertical asymptotes at x=-1 and x=3

what is the equation for this graph?

 

[stapel]

Use what you learned back in algebra! :wink:

The x-intercepts come, naturally, from the zeroes of the numerator. So what factors must you have?

The vertical asymptotes come from the zeroes of the denominator. So what factors must you have?

The horizontal asymptote comes from the relative degrees of the numerator and denominator, and leading coefficients. So what does the horizontal asymptote tell you?

If you get stuck, please reply showing all of your work and reasoning so far. Thank you!

Eliz.

 

[tkhunny]

The x-intercept at x = 1 is a double variety. This increases the degree of the numerator by 1. Do you know this one?

The vertical asymptote at x = -1 requires a denominator fator of EVEN degree - at least two. Why? Do you know this one?

I'm a little concerned about the middle section. My eyes are telling me it isn't symmetric about any vertical axis. I could be wrong, since eyes rarely tell the whole story. In any case, I'd check a point or two on that section before I thought I was done.

Also, any horizontal asymptote just isn't clear from the drawing. It does not look like y = 1, but, again, trusting eyes is not usually a good plan. If it is y = 0, then there are bigger problems. This would require a local maximum somewhere x < -3. Since you are a calculus student, maybe you can figure that out. The max x < 0, with an asymptote y = 0, requires a point of inflection somewhere out there, too.