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Math Help - how to obain the formula and the domain of a parabola

  1. #1
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    how to obain the formula and the domain of a parabola

    Finding the formula associated with each parabola?
    the question is:
    part of a roller coaster is modeled by three parabola - some parts removed to give the diagram below.
    The first parabola from the left meets the second at co-ordinates (7, 14). using this information from the diagram how can i find the formula associated with each parabola and also workout their relevant domains? hint: think of the 3rd parabola is a translation of the first.

    Here's the diagram:
    how to obain the formula and the domain of a parabola-parabola.jpeg
    after looking at the diagram i plugged the (x, y) into the equation y = ax^2 + bx + c. For each point (x, y).
    max(4.5, 20.25)-->y=ax^2+bx+c-->a(4.5)^2+b(4.5)+c
    -->20.25a+4.5b+c=14
    min(12, 1.5)-->y=ax^2+bx+c-->a(12)^2+b(12)+c
    -->144a+12b+c=1.5
    max(19.5, 20.25)-->=ax^2+bx+c-->a(19.5)^2+b(19.5)+
    -->380.25a+19.5b+c=20.25
    is this correct? what am i meant to do with the co-ordinates (7, 14)? and how do i proceed from this step to obtain not just the formula but ALSO the relevant domains?
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  2. #2
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    Re: how to obain the formula and the domain of a parabola

    Quote Originally Posted by wayneB View Post
    Finding the formula associated with each parabola?
    the question is:
    part of a roller coaster is modeled by three parabola - some parts removed to give the diagram below.
    The first parabola from the left meets the second at co-ordinates (7, 14). using this information from the diagram how can i find the formula associated with each parabola and also workout their relevant domains? hint: think of the 3rd parabola is a translation of the first.

    Here's the diagram:
    Click image for larger version. 

Name:	parabola.jpeg 
Views:	5 
Size:	29.7 KB 
ID:	24416
    after looking at the diagram i plugged the (x, y) into the equation y = ax^2 + bx + c. For each point (x, y).
    max(4.5, 20.25)-->y=ax^2+bx+c-->a(4.5)^2+b(4.5)+c
    -->20.25a+4.5b+c=14
    min(12, 1.5)-->y=ax^2+bx+c-->a(12)^2+b(12)+c
    -->144a+12b+c=1.5
    max(19.5, 20.25)-->=ax^2+bx+c-->a(19.5)^2+b(19.5)+
    -->380.25a+19.5b+c=20.25
    is this correct? what am i meant to do with the co-ordinates (7, 14)? and how do i proceed from this step to obtain not just the formula but ALSO the relevant domains?
    1. In general your approach to solve this question is OK:

    If you want to use the equation of a parabola: y = ax^2+bx+c then you need the coordinates of three points of the parabola. You'll get a system of 3 simultaneous equations with 3 variables ((a,b,c)). Since you know 3 points of each parabola this way will lead to a valid solution. (You certainly noticed that (17, 14) is a point on the 2nd or 3rd parabola)

    2. BUT: Since you know the vertex and at least one additional point it would be easier to use the vertex form of the equation of a parabola: If V(x_V, y_V) is the vertex of a parabola it has the equation: y = a(x-x_V)^2+y_V

    3. The equation of the 1st parabola is then: y = a \left(x-\frac92\right)^2+\frac{81}4. Plug in the coordinates (7, 14):

    14 = a \left(7-\frac92\right)^2+\frac{81}4~\implies~a = -1

    Therefore: y = (-1) \left(x-\frac92\right)^2+\frac{81}4

    Expand the bracket and collect like terms: y = -x^2+9x

    4. The two other equations can be obtained in a similar way.
    Last edited by earboth; August 3rd 2012 at 10:39 PM.
    Thanks from wayneB
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    Re: how to obain the formula and the domain of a parabola

    Quote Originally Posted by earboth View Post
    1. In general your approach to solve this question is OK:

    If you want to use the equation of a parabola: y = ax^2+bx+c then you need the coordinates of three points of the parabola. You'll get a system of 3 simultaneous equations with 3 variables ((a,b,c)). Since you know 3 points of each parabola this way will lead to a valid solution. (You certainly noticed that (17, 14) is a point on the 2nd or 3rd parabola)

    2. BUT: Since you know the vertex and at least one additional point it would be easier to use the vertex form of the equation of a parabola: If V(x_V, y_V) is the vertex of a parabola it has the equation: y = a(x-x_V)^2+y_V

    3. The equation of the 1st parabola is then: y = a \left(x-\frac92\right)^2+\frac{81}4. Plug in the coordinates (7, 14):

    14 = a \left(7-\frac92\right)^2+\frac{81}4~\implies~a = -1

    Therefore: y = (-1) \left(x-\frac92\right)^2+\frac{81}4

    Expand the bracket and collect like terms: y = -x^2+9x

    4. The two other equations can be obtained in a similar way.
    could you just explain how you got the -1 in this: [ 14 = a(7-4.5)^2+20.25 ==> x =-1 ]
    and how 9x was obtained in this:
    y = -(x-4.5)^2+20.25
    y = -x^2+9x

    is there a step into obtain a or just pluggin values to obtain it? with the 9x was it just 4.5 x 2 = 9 ???
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  4. #4
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    Re: how to obain the formula and the domain of a parabola

    Quote Originally Posted by wayneB View Post
    could you just explain how you got the -1 in this: [ 14 = a(7-4.5)^2+20.25 ==> x =-1 ]
    and how 9x was obtained in this:
    y = -(x-4.5)^2+20.25
    y = -x^2+9x

    is there a step into obtain a or just pluggin values to obtain it? with the 9x was it just 4.5 x 2 = 9 ???
    1.

    14 = a(7-4.5)^2+20.25~\implies~14=6.25a+20.25~\implies~ \\ -6.25=6.25a~\implies~a=-1

    2.

    y = -(x-4.5)^2+20.25~\implies~y=-(x^2-9x+20.25)+20.25~\implies~ \\ y=-x^2+9x
    Thanks from wayneB
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    Re: how to obain the formula and the domain of a parabola

    Quote Originally Posted by earboth View Post
    1.

    14 = a(7-4.5)^2+20.25~\implies~14=6.25a+20.25~\implies~ \\ -6.25=6.25a~\implies~a=-1

    2.

    y = -(x-4.5)^2+20.25~\implies~y=-(x^2-9x+20.25)+20.25~\implies~ \\ y=-x^2+9x
    Thank You so much for your help!
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  6. #6
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    Re: how to obain the formula and the domain of a parabola

    you've posted this twice ... once in the algebra forum (where it belongs) and now in the trig forum.

    Find the Formula Associated with Each parabola

    DO NOT DOUBLE POST.
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