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Math Help - find intercepts, test for symmetry

  1. #1
    Super Member bigwave's Avatar
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    find intercepts, test for symmetry

    a cardiod microphone pattern is given by the following equation:

    \left{({x}^{2}+{y}^{2}-x\right)}^{2}={x}^{2}+{y}^{2}

    the expanded form of this equation is
    x^4-2 x^3+2 x^2 y^2+x^2-2 x y^2+y^4 = x^2+y^2

    (a) Find the intercepts of the graph of the equation

    to find {x}_{intercepts} I set y = 0

    thus resulting in

    x^4-2x^3=0\Rightarrow x^3\left( x-2 \right)

    so {x}_{intercepts} =  0, 2

    to find {y}_{intercepts} I set x = 0

    thus resulting in

    Y^2=1 so {y}_{intercepts} are \pm 1

    (b) Test for symmetry with respect to the x-axis

    my questions is that i do not how the symmetry is found except by observation from the graph. so symmetry is only on x-axis not on y-axis.



    the books answers were (a) \left( 0,0 \right),\left( 2,0 \right) \left(  0,1\right) \left(0,-1 \right)(b)x=axis symmetry

    but wolfframalpha gave something else

    btw like the upgrades to latex
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by bigwave View Post
    a cardiod microphone pattern is given by the following equation:

    \left{({x}^{2}+{y}^{2}-x\right)}^{2}={x}^{2}+{y}^{2}

    the expanded form of this equation is
    x^4-2 x^3+2 x^2 y^2+x^2-2 x y^2+y^4 = x^2+y^2

    (a) Find the intercepts of the graph of the equation

    to find {x}_{intercepts} I set y = 0

    thus resulting in

    x^4-2x^3=0\Rightarrow x^3\left( x-2 \right)

    so {x}_{intercepts} =  0, 2

    to find {y}_{intercepts} I set x = 0

    thus resulting in

    Y^2=1 so {y}_{intercepts} are \pm 1

    (b) Test for symmetry with respect to the x-axis

    my questions is that i do not how the symmetry is found except by observation from the graph. so symmetry is only on x-axis not on y-axis.



    the books answers were (a) \left( 0,0 \right),\left( 2,0 \right) \left(  0,1\right) \left(0,-1 \right)(b)x=axis symmetry

    but wolfframalpha gave something else

    btw like the upgrades to latex
    There are three typical symmetries looked at...Reflection over the x axis, reflection over the y axis and reflection over the line y = x.

    If the substitution y --> -y gives the same equation, then the equation is symmetric over the x axis.

    If the substitution x --> -x gives the same equation, then the equation is symmetric over the y axis.

    If the substituions x --> y and y --> x (ie switch x and y in the equation) gives the same equation, then the equation is symmetric over the x axis.

    This is a long-winded way to say that I agree with your answer for b).

    -Dan
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  3. #3
    Member eXist's Avatar
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    In order to test for symmetry, check these conditions:

    If f(x) = f(-x), then the graph is symmetric on the Y-axis.

    If f(y) = f(-y), then the graph is symmetric on the X-axis.

    Does this make sense why these two conditions give you symmetry?
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  4. #4
    Super Member bigwave's Avatar
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    this was a little confusing because it appears to be an implicit equation. so doesn't mean we really don't have a function since it fails the vertical line test?
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  5. #5
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by bigwave View Post
    this was a little confusing because it appears to be an implicit equation. so doesn't mean we really don't have a function since it fails the vertical line test?
    You are correct...your equation is not a function. However the symmetry tests still work.

    -Dan
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