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Thread: Graph of a function Question

  1. #1
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    Graph of a function Question

    How can you sketch a graph of the function f that satisfies these conditions:

    f(0) = 0

    f"(x)>0 for x not equal to 0

    lim f'(x) = ∞ , lim f'(x) = -∞
    x →
    0 - .............. x → 0+

    lim f(x) = -, lim f(x)=
    x
    →-∞ ..... x→

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  2. #2
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    Re: Graph of a function Question

    Quote Originally Posted by Roundabout View Post
    How can you sketch a graph of the function f that satisfies these conditions:

    f(0) = 0 ...this means the graph passes through (0, 0)

    f"(x)>0 for x not equal to 0 ...this means that the rate of change of the gradient is positive ie the gradient is increasing ie the graph is concave upward for x not equal to 0

    lim f'(x) = ∞ , lim f'(x) = -∞
    x →
    0 - .............. x → 0+

    ... this means that as x approaches 0 from the left, the gradient gets steeper in a positive sense ie the graph almost becomes vertical. (the y axis (line x=0) will act like an asymptote.
    ... Similarly as x approaches 0 from the right, the gradients becomes more and more steeper in the negative sense.

    lim f(x) = -, lim f(x)=
    x
    →-∞ ..... x→

    ... this means that as x moves further and further to the left (ie gets more and more negative, if you like) y moves further and further down. So the left hand side of the graph will be in the 3rd quadrant (think (neg big number, neg big number)). Opposite as x moves further and further to the right, y gets bigger and bigger, so the graph will end up in the 1st quadrant (think, (pos big number, pos big number)).
    Try to draw the graph of y=f(x) now.


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  3. #3
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    Re: Graph of a function Question

    Thanks for the help. Wouldn't condition 2 (concave up) conflict with condition 4?
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  4. #4
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    Re: Graph of a function Question

    Not necessarily. As x approaches 0+, f(x) must approach infinity. So you have a JU shaped graph (if you can picture that).
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    Re: Graph of a function Question

    MMmm still not quite getting it. Maybe need to sleep on it some.
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    Re: Graph of a function Question

    Starts at the bottom left corner(south west) in Quad 3, scoops up along the left hand side of the y-axis.
    Dot at (0, 0).
    "Parabola" shape on right hand side of y-axis, left side approaches y-axis like an asymptote, right hand side heads up north-east.
    (not a very mathematical explanation). Draw it and you'll see it meets all the criteria.
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  7. #7
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    Re: Graph of a function Question

    So does this look right? Except I drew the left side of the parabola too close to the Y axis...

    Graph of a function Question-graph.jpg
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  8. #8
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    Re: Graph of a function Question

    As x approaches 0+, f(x) must approach infinity.
    This is not necessarily true. In fact, the following specific function is continuous everywhere. Also the graph that you drew can not be a parabola for x > 0, since we need the limit of the slope of f(x) as x approaches $0^+$ to be $-\infty$.

    Last edited by johng; Apr 13th 2017 at 06:31 AM.
    Thanks from math951
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