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Thread: rational function graphs with parabola up and down. How to decide?

  1. #1
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    rational function graphs with parabola up and down. How to decide?

    In the graph f(x)=(x^2-x-6)/(x^2-2x), the interval (0, 2) is a up facing parabola.

    In the graph f(x)=(x^2-x)/(x^2-4), the interval (-2,2), is a down facing parabola.

    Without a calculator, how would I know whether the parabola should face up or down? I don't want to test decimal points because I can't use a calculator.
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  2. #2
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    Re: rational function graphs with parabola up and down. How to decide?

    vertical asymptotes at $x = 0$ and $x = 2$

    analyze the behavior of $f(x)$ on both sides of the asymptotes ...

    $\displaystyle \lim_{x \to 0^-} f(x) = -\infty$

    $\displaystyle \lim_{x \to 0^+} f(x) = \infty$

    $\displaystyle \lim_{x \to 2^-} f(x) = \infty$

    $\displaystyle \lim_{x \to 2^+} f(x) = -\infty$

    btw ... the upward curve shape on the interval $(0,2)$ is not parabolic. If it were, the vertex (min) would be at $x=1$ ... minimum on the interval $(0,2)$ is at $x=6-2\sqrt{6}$
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    Re: rational function graphs with parabola up and down. How to decide?

    I will try this. Thank you.
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