Math Help - Graphing Problem

1. Graphing Problem

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.

2. Hello, vvc531!

This is a very messy problem . . .

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.

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

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

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

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

Long division: . $\frac{ax^2 + bx + c}{x+2} \;=\;ax + (b-2a) + \frac{4a-2b + c}{x+2} \;\;\to\;\;-x + 3$

Hence: . $a \:=\:-1$
. . . . . . $b-2a \:=\:3 \quad\Rightarrow\quad b \:=\:1$
. . . . . . $c \:=\:\text{any constant}$

Let $c = 0$ and one function is: . $f(x) \;=\;\frac{x-x^2}{x+2}$

And the graph looks something like this:

Code:
    .              *.       |
.*            .       |
.*         *.       |
. *       .       |
. *   * .       |
.  *  .       |
.   .       |
. .       |
.       |
. .     |
.   .   |
.  *  . |
. *   * .3
.       * .
.*      |  *.
.       |    *. 3
------------------.*------+-------.------
-2       |        *.
.       |           .
.       |

Um, my scale is off.

The intercepts should be: .(0, 0) and (1, 0)

I hope you can modify the graph.