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Question 346540: What is the focus and the directed line using the parabola function of y=x^2-8x-9 thank you!
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! 
First we must transform the equation into a more useful form. For the parabola, with the  term, the form we want is:
An equation in this form is a vertically-oriented parabola with a vertex at (h, k) and the (vertical) distance from the vertex to the focus is "p".
To get your equation in this form, we start by "completing the square". There are different ways of doing this. The way I like is to start by isolating the terms with the variable whose square I am trying to complete. In this case, I am completing the square for x (because of the ). So I want the x terms on one side and everthing else on the other side. So I'll start by adding 9 to each side:
Now we figure out what constant we need to to make the right side a perfect quare trinomial. We take half the coefficient of x, which is -8, and square it. Half of -8 is -4 and -4 squared is 16. The constant we want is 16 so that is what we will now add to each side:
The right side is now a perfect square. It is  . (The -4 comes from half of the coefficient of x.) Now we just have to fix up the left side so it matches the desired form. We want y-k and we have y+25. Fortunately additions can be rewritten as subtractions of the opposite: y+25 = y - (-25), Now we have:
Last of all we need something for the "4p" in the desired form. We can always factor out a 1!
This is the equation we need. The "h" is 4, the "k" is -25 and 4p = 1 (which makes p = 1/4). This makes the vertex of the parabola (4, -25). Since this is a vertical parabola (because of the ) so the focus is "p" units up from the vertex and the directrix is a horizontal line "p" units down from the vertex.
So the focus is (4,  ) or (4,  ) and the directrix is the line: y =  or y = 
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