Graphing Problem

Feb 2009
13
0
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.
 

Soroban

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May 2006
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Hello, vvc531!

This is a very messy problem . . .


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

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

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


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


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.