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Question 1201701: The equation of a parabola is 12 y = ( x − 1 ) 2 − 48 12y=(x-1)2-48 . Identify the vertex, focus, and directrix of the parabola.
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
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NOTE for future reference: the symbol "^" (shift-6) is commonly used to represent exponents. So you can write the equation for this problem as
12y=(x-1)^2-48
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The x term is squared, so the graph opens up or down. The general vertex form of the equation of a parabola I prefer to use is this:
Note many references will show this equation in different equivalent forms; and different students have different preferences on which form to use. Some common equivalent forms are
 [puts only "y" on the left]
 [keeps the "4p" on the left so it is not a fraction]
In any of those forms, the vertex is (h,k); p is the directed distance (i.e., can be negative) from the directrix to the vertex and from the vertex to the focus.
Put the equation in your example in this form:
The vertex is (1,-4) and p is 3.
The directrix is p = 3 units below the vertex, at y = -7.
The focus is p = 3 units above the vertex, at (1,-1).
ANSWERS:
vertex (1,-4)
focus (1,-1)
directrix y = -7
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