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Math Help - Graph of the equation is symmetric

  1. #1
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    Thumbs down Graph of the equation is symmetric

    I need help solving this problem, can anyone do a step by step guide on how to do this?

    Determine whether the graph of the equation is symmetric with respect to the line Y = -X

    1) Y = 2X - 1
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  2. #2
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    One way of checking it is by slopes, because both x = -y and y = 2x +1 are straight lines only.
    If y = 2x +1 were symmetrical with respect to y = -x, then y = 2x +1 should be perpendicular to y = -x. If not, then there is no symmetry.

    y = -x
    slope, m1 = -1

    y = 2x +1
    slope, m2 = 2

    To be perpendicular, their slopes must be the negative reciprocals.
    m2 = -1/m1
    2 = -1 / -1
    2 = 1
    False, so the two lines are not perpendicular, and so y = 2x +1 is not symmetrical wirh respect to y = -x.

    -------------------------
    Another way.

    y = -x -------------axis of symmetry.

    y = 2x +1 --------(i)

    If (i) is symmetrical, when we swap the x and y into (i), we should get an equation similar or equal to (i).

    -x = 2(-y) +1
    -x = -2y +1
    2y = x +1
    y = (1/2)(x +1) -------not the same as (i), so, no symmetry.
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  3. #3
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    Quote Originally Posted by ticbol View Post
    y = -x -------------axis of symmetry.

    y = 2x +1 --------(i)

    If (i) is symmetrical, when we swap the x and y into (i), we should get an equation similar or equal to (i).

    -x = 2(-y) +1
    -x = -2y +1
    How come the -x remained as a -x. While the 2y turned into a -2y and the -1 turned into a 1?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Brazuca View Post
    How come the -x remained as a -x. While the 2y turned into a -2y and the -1 turned into a 1?
    what are you talking about? he replaced y with -x and x with -y and solved for y. that's it.
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    what are you talking about? he replaced y with -x and x with -y and solved for y. that's it.
    I don't understand.

    If Y = -X then wouldn't the -X have to turn into a X for the Y to turn into a -Y?

    -x = 2(-y) +1
    -x = -2y +1

    Wouldn't the problem have to turn into this?
    -x = 2y + 1
    x = 2(-y) +1
    x = -2y + 1
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Brazuca View Post
    I don't understand.
    we have
    y = -x which is also saying that x = -y, this is line (1)

    then we are given y = 2x - 1 ........line (2)

    to find if y = 2x - 1 is symmetric with respect to line one, plug in y = -x and x = -y as line (1) directed. if we simplify and get the original formula for line (2), then line (2) is symmetric with respect to line (1)

    {\color {red}y } = 2 {\color {blue}x} - 1

    Plug in y = -x and x = -y, we get:

    {\color {red}-x } = 2({\color{blue}-y}) - 1

    solving for y, we get:

    y = \frac 12x - \frac 12

    which is not the original formula for line (2), so line (2) is NOT symmetric with respect to line (1)
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  7. #7
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    Quote Originally Posted by Jhevon View Post
    we have
    y = -x which is also saying that x = -y, this is line (1)
    Thanks, I had to read everything you posted step by step to understand that y = -x then x = -y.

    I always looked at it as y = -x then -x = y. I was not switching the signs.
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  8. #8
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Brazuca View Post
    Thanks, I had to read everything you posted step by step to understand that y = -x then x = -y.

    I always looked at it as y = -x then -x = y. I was not switching the signs.
    if you multiply both sides of the equation y = -x by -1 you get -y = x
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