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Math Help - Equation of parabola joined smoothly to the line y=2x at a point.

  1. #1
    Member Mr Rayon's Avatar
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    Equation of parabola joined smoothly to the line y=2x at a point.

    Find the equation of the parabola joined smoothly to the straight line y=2x at the point P(2,4) as shown below. The parabola passes through the point(5,1).

    y = a(x - b)^2 + c
    y = a(x - 2)^2 + 4
    1 = a(5 - 2)^2 + 4
    1 = a(3)^2 + 4
    1 = 9a + 4
    a = \frac{-1}{3}

    y = \frac{-1}{3}(x - 2)^2 + 4
    Attached Thumbnails Attached Thumbnails Equation of parabola joined smoothly to the line y=2x at a point.-juice.jpg  
    Last edited by mr fantastic; June 20th 2010 at 06:19 PM. Reason: Merged posts from different threads.
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  2. #2
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    The solution here:

    Spoiler:

    y=ax^2+bx+c

    1=25a+5b+c

    y'(x)=2ax+b

    2=4a+b

    4=4a+2b+c
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  3. #3
    Member Mr Rayon's Avatar
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    Is this right?
    a = -9/36
    b = 3/4
    c = -9

    Can't be bothered showing working out.
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  4. #4
    MHF Contributor Unknown008's Avatar
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    Firstly, you are given two points, (2, 4) and (5, 1), and you are given the gradient at the point (2, 4) which is 2.

    Let's say that your equation has the form:

    y = ax^2 + bx + c.

    Plug in the coordinates to have a pair of equations:

    (4) = a(2)^2 + b(2) + c --> 4 = 4a + 2b + c -I

    (1) = a(5)^2 + b(5) + c --> 1 = 25a + 5b + c -II

    Now the gradient on the curve is given by the derivative of the curve.

    y' = 2ax + b

    Bud in the value of x and the value of y':

    (2) = 2a(2) + b

    2 = 4a + b -III

    Now, eliminate 'c' from the first two equations (II - I)to get:

    -3 = 21a + 3b

    Reduce to:

    -1 = 7a + b -IV

    Now, use simultaneous solving to get the values of a and b from equations III and IV.

    you should get a = -1, b = 6 and c = -4
    Last edited by Unknown008; June 20th 2010 at 10:05 AM.
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  5. #5
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    a = -9/36
    b = 3/4
    are not satisfying 4a+b=2

    -9/4+3/4 \neq 2
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  6. #6
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    mr fantastic's Avatar
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    Quote Originally Posted by Mr Rayon View Post
    Is this right?
    a = -9/36
    b = 3/4
    c = -9

    Can't be bothered showing working out.
    And yet you want people to be bothered working the problem from scratch to check these answers.

    If you want help, it is in your own interests to show your work so that people can more easily help you. eg. You might have made a mistake in your very first line of working, this could then be easily pointed out and people wouldn't have to waste time doing their own calculations from scratch.
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