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Math Help - Three Graphs Include y = x

  1. #1
    Senior Member nycmath's Avatar
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    Three Graphs Include y = x

    I want to know why the following three graphs include the line y = x?

    (1) g(x) = x + [4/(x^2+1)

    (2) f(x) = (x^2+1)/x

    (3) y = (x^2-6x+12)/(x-4)

    I am trying to learn curve sketching from my single variable calculus textbook. I am not asking you to graph all three. I just want to know why the above functions include the line y = x as part of the graph.
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  2. #2
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    Re: Three Graphs Include y = x

    I'm not entirely sure what you mean by "includes y=x as part of the graph" but from what I think you mean.

    1) g(x)=x+\frac{4}{x^2+1}

    this function tends towards x as the second term tends towards zero so asymptotically g(x)=x

    2) f(x)=\frac{x^2+1}{x}=x+\frac{1}{x}

    this function also tends towards x for the same reason, the second term tends to 0 as x grows.

    3) y=\frac{x^2-6x+12}{x-4}=(x-2)+\frac{4}{x-4}

    y tends to (x-2) here as x grows so once again asymptotically y looks like x albeit offset by the constant 2.

    Is this what you are after?
    Last edited by romsek; January 24th 2014 at 05:01 PM.
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    u

    Hello, nycmath!

    I want to know why the following three graphs include the line y = x.

    (1)\;g(x) \:=\: x + \frac{4}{x^2+1}

    (2)\;f(x) \:=\:\frac{x^2+1}{x}

    (3)\;y \:=\;\frac{x^2-6x+12}{x-4}

    That line isn't a part of the graph.
    It is an asymptote; the graph approaches that slanted line.


    Consider: . (1)\;y \:=\:x + \frac{4}{x^2+1}

    As x\to\infty, the fraction goes to 0.
    Hence, the graph gets closer and closer to y = x.

    In Quadrant 1, the graph looks like this:
    Code:
          |
          |*
          |
          |                   .
          | *              *.
          |              *.
          |  *         *.
          |   *     * .
          |     *   .
          |       .
          |     .
          |   .
          | .
      - - + - - - - - - - - - - - -
          |
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  4. #4
    Senior Member nycmath's Avatar
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    Re: Three Graphs Include y = x

    I will practice more curve sketching using calculus 1 techniques. Soroban, check out your inbox. Have a great weekend guys.
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    Re: Three Graphs Include y = x

    Quote Originally Posted by nycmath View Post
    I want to know why the following three graphs include the line y = x?
    (1) g(x) = x + [4/(x^2+1)
    (2) f(x) = (x^2+1)/x
    (3) y = (x^2-6x+12)/(x-4)
    For the life of me. I cannot understand why you do not use free on line tools?
    Look at this.

    You can change the input there to see other graphs.
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  6. #6
    Senior Member nycmath's Avatar
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    Re: Three Graphs Include y = x

    Plato: I am trying to learn curve sketching without the use of technology. Thanks for the link.
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  7. #7
    Senior Member nycmath's Avatar
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    Re: Three Graphs Include y = x

    I use wolframalpha.com all the time. However, it's a lot more fun trying to graph these functions using curve sketching techniques from calculus 1 textbooks.
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    Re: Three Graphs Include y = x

    Quote Originally Posted by nycmath View Post
    Plato: I am trying to learn curve sketching without the use of technology. Thanks for the link.
    O.K. But why not goto WFA and look at the graph? As you look at it, ask yourself "why does that work?"

    Does any of that actually work?

    If you can answer that question, then pass it on.
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