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Math Help - Relationship between apex and derivative

  1. #1
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    Relationship between apex and derivative

    Does somebody know if there's relationship between apex and derivative ?
    If there's formula y=ax^2+bx+c then the x apex is -b/2a and y apex is y(x). But can I get the same result by using function's derivative ?

    I tried to determine a and b according to the condition that parabola's y=a+bx-x^2 apex point would be (1;2)
    I assumed that at the apex the slop is 0 and so did some substitutions:
    firstly first derivative is: y'=0+b-2x
    0=b(1)-2
    b=2
    y=2 so 2=a+2-1
    2=a+1
    a=1

    It appears to be correct but the way I am used to use for finding apexes don't work with the formula y=a+bx-x^2

    Any ideas ?
    Last edited by totalnewbie; August 18th 2005 at 09:25 AM.
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  2. #2
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    So, what is your problem? What don't you understand? You got it correctly [except "0 = b(1) -2". That should have been 0 = b -2(1)], so ....?

    You used derivatives--yes, the derivative is correct, and yes, at the "apex" or vertex, the slope of the tangent line, or the 1st derivative, is zero.
    So, what is wrong?

    -------------------
    "...but the way I am used to use for finding apexes don't work with the formula y=a+bx-x^2."

    And what is that way?

    The "x = -b/2a" way?

    y = a +bx -x^2
    y' = b -2x
    y' = 0
    0 = b -2x
    2x = b
    x = b/2 --------the x-coordinate of the apex/vertex of y = a +bx -x^2.

    Is that wrong?
    Is this the reason why it does not work with y = a +bx -x^2 ?

    y = a +bx -x^2
    Rewriting that,
    y = -x^2 +bx +a ----(1)
    At the vertex,
    x = "-b/2a"
    Here, in (1), "b" is +b; and "a" is -1, so,
    x = -(+b)/[2(-1)]
    x = -b/-2
    x = b/2 ----------is that the same as the x = b/2 above?
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  3. #3
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    Quote Originally Posted by ticbol
    y = a +bx -x^2
    y' = b -2x
    y' = 0
    0 = b -2x
    2x = b
    x = b/2 --------the x-coordinate of the apex/vertex of y = a +bx -x^2.
    That's the one I wanted to know. Thanks for that.
    You shouldn't write so long explanation. The one which is above on quota is enough to understand.
    Last edited by totalnewbie; August 19th 2005 at 01:59 AM.
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  4. #4
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    You have a problem with me then.
    I always give long explanations.
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