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Downward parabola: value of p = 1/a?

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  1. Feb 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Hello!
    I am repeating conics, and have stumbled upon this paragraph on purplemath.com
    "
    Then a = –1/9. With a being the leading coefficient from the regular quadratic equation y = ax^2 + bx + c, I also know that the value of 1/a is the same as the value of 4p, so 1/(–1/9) = –9 = 4p, and thus p = –9/4. "

    I am not aware of the rule that states that 4p equals 1/a for a parabola that has a vertex other than
    (0, 0), which is the case here. The author refers to some general formula of parabola y = ax^2 + bx + c.
    If vertex is at (0, 0 ), then the formula is x^2 = 4py, hence y = 1/4p * x^2, where a = 1/4p, hence
    a = 1/4p. But in the given example, the vertex is not at (0, 0) - it's at (0, 25). Why a = 1/4p if the
    vertex is not (0, 0)?

    I would be grateful for explanation.

    Thank you!

    2. Relevant equations


    3. The attempt at a solution
     
  2. jcsd
  3. Feb 29, 2016 #2
    This result can easily be shown for any parabola, not just one with vertex at the orgin e.g in this section of a Wikipedia page.
     
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