[Math] Writing a rational function with given characteristics

algebra-precalculus

I've got two homework question that have me stumped. Both lack an x-intercept, and the second one throws an oblique asymptote into the mix.

Problem one provides the following characteristics:

Vertical asymptotes at $x=-2$, and $x=5$, Hole in graph at $x=0$, Horizontal asymptote at $y=3$

What has me stumped is what am I supposed to do with the numerator? All the previous question had an x-intercept.

Here's what I have so far:
$\dfrac{x}{x} \cdot \dfrac{3(???)}{(x+2)(x-5)}$

Problem two also does not provide an x-intercept. This problem also has an oblique asymptote that I don't know how to handle.

Here are the characteristics:
Vertical asymptotes at $x=2$ and $x=-4$, Oblique asymptote at $y=2x$

Any help would be greatly appreciated!

Best Answer

  1. The function has to have $\lim_{x\rightarrow\pm\infty}=3$ . To do this, the numerator must be a polynomial of the same degree as the denominator (so neither overpowers the other), with a $3$ as the coefficient of the largest term. Basically a number of functions will work, such as:

$$\frac{3x(x^2+1)}{x(x+2)(x+5)}$$

Both cubics, with a $3x^3$ on top and an $x^3$ on the bottom. So as $|x|$ increases the smaller terms ($x^2$,etc.) will drop away to leave $3$. There are no $x$ intercepts, since $x^2+1\neq 0$ for any $x$.

For the oblique asymptote the idea is the same, but now the numerator should be larger than the denominator, so that the two largest terms divide to give $2x$. Try it yourself, and I'll edit this answer if you're still stuck.

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