Find A Rational Function

[mathdad]

In Section 5.3, Sullivan breaks down rational functions in specific detail. In the last few pages of the section, he introduces the idea of forming a rational function from a given graph. Not too many questions, maybe 4, in terms of this idea.

So, a graph is given. The graph shows vertical and horizontal asymptotes, x-intercepts and y-intercepts. I need to use this information to form a rational function in the form f(x) = [a•p(x)/q(x)], where a is constant.

I know:

A. Vertical asymptotes reveal the denominator(s).
B. I know the x-intercepts reveal the numerator(s).
C. I know the y-intercept is used to find the value of the constant a.

I DO NOT know what the horizontal asymptote(s) is used for. I have not encountered a graph with oblique asymptotes. Like I said, not too many questions by Sullivan in this regard.

QUESTION:

What is the easiest way to form a rational function from a given graph? See the picture. Form a rational function.

 

[Dr.Peterson]

Basically,

• Use the x-intercepts (and multiplicities) to find the factors of the numerator.
• Use the vertical asymptotes (and multiplicities) to find the factors of the denominator.
• Use "holes", if any, to add factors to both numerator and denominator.
• Use the y-intercept to find the vertical scale (your "a").
• Use the horizontal (or oblique) asymptote to check the vertical scale when you are finished.

That is, a vertical or oblique asymptote isn't independent of the rest; it is whatever it turns out to be, so a problem could be inconsistent if you picked it at random. You could instead use this to find "a" and then use the y-intercept to check, but the latter is usually easier to use, and the former to check.

 

[harpazo]

Thank you.