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Math Help - Minor clarification on modulus graphs

  1. #1
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    Minor clarification on modulus graphs

    If the equation of a curve is y=(x-4)^2+(-1). It shows that the curve has a minimum point of (4,-1).

    Now, if I have a modulus of this equation, that is |(x-4)^2+(-1)|, the turning point would be (4,1). Is the turning point a maximum point or minimum point now?
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  2. #2
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    Quote Originally Posted by Punch View Post
    If the equation of a curve is y=(x-4)^2+(-1). It shows that the curve has a minimum point of (4,-1).

    Now, if I have a modulus of this equation, that is |(x-4)^2+(-1)|, the turning point would be (4,1). Is the turning point a maximum point or minimum point now?
    Suppose x is close to 4, say x= 3.9 or x= 4.1. If x= 3.9 |(x-4)^2- 1|= |(3.9- 4)^2- 1[tex]= |(-.1)^2- 1|= |.01- 1|= |-.99|= .99 which is less than 1. If x= 4.1, |(x-4)^2- 1|= |(4.1-4)^2- 1| = |(.1)^2- 1|= |.01- 1|= |-.99|= .99 which is also less than 1. Now, do YOU think (4,1) is a maximum or minimum point?

    There are, in fact, three turning points for that graph, (3, 0), (4, 1), and (5, 0).
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  3. #3
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    Well, I am just speaking in general, looks like you are going really in-depth. The turning point I am referring to is (4,1), just unclear about whether it would be called the maximum point or minimum point...
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  4. #4
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    Hello, Punch!

    The equation of a curve is y\:=\:(x-4)^2+(-1).
    It shows that the curve has a minimum point of (4,-1).

    Now, if I have a modulus of this equation, that is: y \:=\:|(x-4)^2+(-1)|,
    . . the turning point would be (4,1).

    Is the turning point a maximum point or minimum point now?
    Visualize their graphs . . .


    The graph of the parabola looks like this:
    Code:
          |
          |  *                 *
          |
          |
          |   *               *
          |
        --+----*-------------*-----
          |     *           *
          |       *       *
          |           *
          |

    There is an absolute minimum at (4, -1).
    There is no maximum.



    With absolute values, any point below the x-axis
    . . is reflected upward.

    The graph of the modulus function is:


    Code:
          |
          |  *                 *
          |
          |           *
          |   *   *       *   *
          |     *           *
        --+----*-------------*-----
          |    3             5
          |

    It has absolute minimums at (3,0) and (5,0).
    . . and a relative (local) maximum at (4,1).

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  5. #5
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    Quote Originally Posted by Soroban View Post
    Hello, Punch!

    Visualize their graphs . . .


    The graph of the parabola looks like this:
    Code:
          |
          |  *                 *
          |
          |
          |   *               *
          |
        --+----*-------------*-----
          |     *           *
          |       *       *
          |           *
          |
    There is an absolute minimum at (4, -1).
    There is no maximum.



    With absolute values, any point below the x-axis
    . . is reflected upward.

    The graph of the modulus function is:

    Code:
          |
          |  *                 *
          |
          |           *
          |   *   *       *   *
          |     *           *
        --+----*-------------*-----
          |    3             5
          |
    It has absolute minimums at (3,0) and (5,0).
    . . and a relative (local) maximum at (4,1).
    Thanks Soroban, this was what I was asking for!
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