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Math Help - Symmetry

  1. #1
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    Symmetry

    Hello,

    I am having a difficult time with this so could someone please explain and show me how to exactly do this:

    Determine whether the graph of f(x)=x+6x is symmetric with respect to the line y=x the line y=-x and/or the origin.


    Thanks,

    Nick
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  2. #2
    Bar0n janvdl's Avatar
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    I've drawn everything for you, see if you can figure anything out
    Attached Thumbnails Attached Thumbnails Symmetry-graph.jpg  
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  3. #3
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    That is great that you gave me a graph, but I really need to know HOW to do it. Someone suggested an x and y table, but I just need help setting things up. I am really stuck, please help me.
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  4. #4
    MHF Contributor

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    I dont think that quite answers the question!

    The point that is "symmetric" to (4, 0) in the line y= x is the point (0, 4) (i.e. a line from (4,0) to (0,4) is perpendicular to y= x and those point are equidistant from the line). In general, the point that is "symmetric" to (x,y) in the line y= x is the point (y, x). Just "swap" x and y coordinates.

    To determine if the graph of y= x+6x is "symmetric with respect to the line y= x", swap x and y to get x= y+ 6y. Is that the same graph?

    To determine if it is "symmetric with respect to the line y= -x", replace y with -x and x with -y: -x= (-y)+ 6(-y). Is that the same graph?

    Finally, to determine "if it is symmetric with respect to the origin", replace x by -x, y by -y: -y= (-x)+ 6(-x). Is that the same graph?
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  5. #5
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    Hello, Nick!

    There are tests for these symmetries . . .


    Determine whether the graph of f(x)\:=\:x^3+6x is symmetric
    with respect to the line y=x, the line y=-x and/or the origin.
    Symmetry to the origin

    \text{Replace }x\text{ with }\text{-}x\text{ and }y\text{ with }\text{-}y.
    \text{If the result is the original equation, the graph is symmetric to the origin.}

    We have: . y \:=\:x^3 + 6x

    Replace: . \text{-}y \:=\:(\text{-}x)^3 + 6(\text{-}x)\quad\Rightarrow\quad \text{-}y \:=\:\text{-}x^3 - 6x
    . . Multiply by -1: . y \:=\:x^3 + 6x . . . the original equation!
    The graph is symmetric to the origin.



    Symmetry to y = x

    \text{Replace }x\text{ with }y\text{ and }y\text{ with }x.
    \text{If the result is the original equation, the graph is symmetric to }y = x.

    Replace: . x \:=\:y^3 + 6y
    . . This cannot be made equal to the original equation.
    The graph is not symmetric to y = x.



    Symmetry to y = -x

    \text{Replace }x\text{ with }y\text{ and }y\text{ with }\text{-}x.
    \text{If the result is the original equation, the graph is symmetric to }y = -x.

    Replace: .  \text{-}x \:=\:y^3 + 6y
    . . This cannot be made to equal the orignal equation.
    The graph is not symmetric to y = -x.

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  6. #6
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    Asymptotes

    Thanks, that helped me a lot.
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