Suppose that a rational function approaches the lines x=-2 and y=3-x asymptotically. Sketch a graph of this function and write an equation that describes this function.

Suppose that a rational function approaches the lines \(\displaystyle x=-2\) and \(\displaystyle y\:=\:3-x\) asymptotically.
Sketch a graph of this function and write an equation that describes this function.

The vertical asymptote is simple: we need \(\displaystyle (x+2)\) in the denominator.

Now we want a function: .\(\displaystyle f(x) \;=\;\frac{p(x)}{x+2} \) .which approaches .\(\displaystyle y \:=\:-x+3\) .as \(\displaystyle x \to\infty.\)

Hence, the numerator \(\displaystyle p(x)\) must be a quadratic.

We want: .\(\displaystyle \lim_{x\to\infty}\frac{ax^2+bx+c}{x+2} \;=\;-x + 3\)