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Math Help - How do you know the shape of a function by looking at the equation?

  1. #1
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    How do you know the shape of a function by looking at the equation?

    F(x,y) = 40x - x2 + 15y - 4y2

    How do I know what this function is supposed to look like without drawing it out on paper?


    Any help would be greatly appreciated
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  2. #2
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    Re: How do you know the shape of a function by looking at the equation?

    Hey mathbeginner97.

    Usually what happens is that you calculate the when the first derivatives are zero and when the function is also zero. In other words find when f(x) = 0 and when f'(x) = 0.

    Once you do that you draw a smoioth line that goes through these points to get an idea of what the curve looks like.

    Have you done calculus yet (i.e. differentiation)?
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  3. #3
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    Re: How do you know the shape of a function by looking at the equation?

    Yes I have, but I haven't done multivariable calculus so I don't know what's going on
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  4. #4
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    Re: How do you know the shape of a function by looking at the equation?

    For this particular function, F(x,y) = 40x - x^2 + 15y - 4y^2, complete the square:
    F(x, y)= -(x^2- 40x)- 4(y^2- (15/4)y)= -(x^2- 40x+ 400- 400)- 4(y^2- (15/4)y+ (225/64)- (225/64))
    = -(x- 20)^2+ 400- 4(y- 15/8)^2+ 225/8= 3425/8 -(x- 20)^2- 4(y- 15/8)^2

    That's a "paraboloid" with vertex at (20, 15/8, 3425/8) and axis the line x= 20, y= 15/8, z= t.

    Answering your question more generally is exactly what "multi-variable calculus" is for.
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