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Math Help - Graph

  1. #1
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    Graph

    I need hints on the following questions:


    1. Sketch the curve (y2-2)2+(x2-2)2=2. What does it look like?

    I got 4 circles which kind of merge together where they meet.


    2. kx^4=x^3-x Find the real roots when k=0. Sketch the graph when k is small and then when k is large, and find approximations of the real roots in both cases. When else does x have 3 real roots?

    I don't understand why the graph has 3 roots, even when k is not 0?


    3. sketch x^x

    The graph is a bit complicated, and broken down in the negative x values. I'm not sure how to draw it when x = 0, on the calculator it doesn't show it as an asymptote or anything, is it just supposed to end there or what? Also, I dont understand how to work out the fluctuations between positive and negative values of y for the negative values of x.


    4. Sketch f(x) = (x - R(x))2, where R(x) is x rounded up or down in the usual way. then sketch g(x) = f(1/x)

    For f(x) I got like multiple humps between the integer, with maximum at half way between 2 integers of 1/4, and then 0 at the integers. However, I have no idea for g(x), it doesn't seem to follow any pattern.
    Last edited by Aquafina; November 1st 2009 at 10:04 PM.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello Aquafina
    Quote Originally Posted by Aquafina View Post
    I need hints on the following questions:


    1. Sketch the curve (y2-2)2+(x2-2)2=2. What does it look like?

    I got 4 circles which kind of merge together where they meet.
    Note that we have just even powers of x and y, so the graph is symmetrical about both axes. If we re-arrange:

    y^2=2\pm\sqrt{2-(x^2-2)^2}

    Note that whenever \sqrt{2-(x^2-2)^2} is real, 2-\sqrt{2-(x^2-2)^2} \ge 0 so y is real.

    Then \sqrt{2-(x^2-2)^2} is real whenever (x^2-2)^2\le 2

    i.e. \sqrt{2-\sqrt2}\le x \le \sqrt{2+\sqrt2} or -\sqrt{2+\sqrt2}\le x \le -\sqrt{2-\sqrt2}.

    These give approximate values of 0.76 \le x \le 1.85 and -1.85\le x \le - 0.76.

    By the symmetry of the equation, y has the same ranges of values.

    So the graph consists of four non-overlapping loops as in the attached sketch.

    Grandad
    Attached Thumbnails Attached Thumbnails Graph-untitled.jpg  
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  3. #3
    Senior Member pacman's Avatar
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    3. y = x^x. Details are not included, these are the progressive graphs, try to fill in the details
    Attached Thumbnails Attached Thumbnails Graph-2.gif   Graph-3.gif   Graph-5.gif  
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