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Math Help - Determining Lines of Symmetry from Scratch?

  1. #1
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    Determining Lines of Symmetry from Scratch?

    Hi all ---

    When I tried the following question in the black oval --- the answer is in red --- I was completely stuck on the logic/thought process/intuition in the first part. I'm fine with the algebra --- it's NOT what's troubling me for the past hours.

    I know how to check for symmetry about the origin and y-axis. A function, f(x), is symmetric about the origin if  f(-x) = -f(x) and about y-axis if  f(-x) = f(x).

    But here, how on earth would you even know to check for symmetry about  y = \pm x ? How would you even suspect or be aware of this in the first place?

    My work: For  xy = v \Longleftrightarrow y = \frac{v}{x}, I sort of see this because this is just the reciprocal function with the constant  v.
    So here, I'd expect symmetry about  y = \pm x because I've seen the graph of the reciprocal function before. But I'm completely lost for x^4 + y^4 = u?

    Also, what would lead you to check for symmetry about x- and y-axis here? Question doesn't give any information that would lead you to do this?

    In general, if I see this kind of question that only asks me to state all lines of symmetry (WITHOUT telling me what the lines are), how would I know
    --- which lines I should check for symmetry?
    --- how many lines there are?

    Maybe because I don't understand it, but the answer looks like a bit of guesswork to me...

    Thank you ---

    Last edited by mathminor827; May 30th 2012 at 10:19 AM.
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  2. #2
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    Re: Determining Lines of Symmetry from Scratch?

    The fact that the exponents on x and y are even so that (-x)^4= x^4 and (-y)^4= y^4 would lead me to say that the graph is symmetric about the axes: the point "symmetric" to (x, y) about the y-axis is (-x, y) and about the x-axis is (x, -y). The fact that the formula is "symmetric" with respect to x and y: replacing x with y and y with x changes x^4+ y^4= u to (y)^4+ (x^4)= u which is the same equation, tells me that it is symmetric about the lines y= x and y= -x.
    Last edited by HallsofIvy; May 30th 2012 at 10:40 AM.
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  3. #3
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    Re: Determining Lines of Symmetry from Scratch?

    Hi HallsofIvy ---

    Thanks a lot for your answer. It really helped with why one'd think  y = \pm x and  x, y axes give symmetry for the functions.

    Just one last question that was maybe unclear in my post --- How would you know/infer that there are no other lines of symmetry to check or test for either curve?
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  4. #4
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    Re: Determining Lines of Symmetry from Scratch?

    After thinking about this for the past week, I still do not see how and why there are no other lines of symmetry? Has someone gotten farther than me?
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  5. #5
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    Re: Determining Lines of Symmetry from Scratch?

    Hi all again --- I should point out that the red box is the solution, so we are NOT told or hinted that both graphs are symmetric about  y = \pm x and  x = 0, y = 0 .

    After two more weeks of banging my head on this problem, I still do not see how you can tell or guess that there are no other lines of symmetry? Does anyone have more insight?

    Thank you ---
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