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Thread: Parabola equations and line through parabola equations

  1. #1
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    Parabola equations and line through parabola equations

    Find the formula for the parabola that passes through the points (2,2) and (5,7) whose graph is given below:
    Parabola equations and line through parabola equations-thom-3397-setr-t2prob5image1.png

    Since (2,2) is the vertex, could I put this in vertex form y= (x-h)2+k for (x-2)2+2? I'm not sure how to incorporate the (5,7) point in though.

    and

    Finding the equation for a line that goes through a parabola, with the parabola's equation being y = x2 + 2
    Parabola equations and line through parabola equations-thom-354-setr-t2prob6image1.png
    I'm not sure where to even begin with this one. All the variables are throwing me off
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  2. #2
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    Re: Parabola equations and line through parabola equations

    1. $y=a(x-2)^2+2$ ... sub in 5 for x and 7 for y and solve for $a$.

    2. From the diagram, looks like point Q has x-coordinate 3 and point P has y-coordinate 18. If that's the case, then point Q has coordinates (3,11) and point P has coordinates (4,18) ... you should be able to find the equation between two points, right?
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    Re: Parabola equations and line through parabola equations

    Well, I'm not the best at Math either, but I'll try to help you out.

    For the second question, I'm assuming that you just need any line that passes through the parabola with equation y = x^2 + 2. The form for an equation of a parabola is y = ax^2 + bx + c. y = x^2 + 2 is y = 1x^2 + 2.

    The x^2 part tells you that for each value of x, y will be x^2 that, so if x=4, y would equal 16. However, the + 2 part tells us that for whatever value of x^2, we need to add 2 to it. So when x=4, y would actually equal 18. Now all you have to do is pick an equation for a line that would pass through this.
    y = x + 2 would pass through the vertex, and y = 2x + 4 would also pass through the parabola. Any other equation the puts a line through the parabola would work as well I think.


    For the first question, you would only need (5, 7) to find 'a', given the vertex form y = a(x-h)2+k. 'a' is kind of like the slope of the parabola, and you find it in pretty much the same way you find the slope of a line:

    1. Pick any known point on the parabola (in this case, pick the vertex)
    2. Move over 1 unit to the right (if you start at (2,2), go to (3,2)) and see what the y value is for that point on the parabola (3,y).
    3. Since you don't know the point 1 unit to the right, just solve for a ;p (please look up how to solve for a, I'd say with some research you could find a good answer)
    4. Input the point (5,7) into the equation y = a(x-h)^2+k.
    5. That would be: 7 = a(5-2)^2+2 --> 7 = a(3)^2+2 --> 7 = 9a + 2 --> 7-2 = 9a+2-2 --> 5 = 9a --> a = 5/9
    6. Check your answer by substituting 'a' back into the equation.

    Hope I helped, I strongly encourage you to look this subject up on your own though, as there may be a quicker/easier way to solve this.
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