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Math Help - Graphing Problem

  1. #1
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    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.
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  2. #2
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    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.

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