document.write( "Question 346540: What is the focus and the directed line using the parabola function of y=x^2-8x-9 thank you! \n" ); document.write( "

Algebra.Com's Answer #247867 by jsmallt9(3758) $\"\"$ $\"About$

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$\"y+=+x%5E2+-8x+-9\"$
\n" ); document.write( "First we must transform the equation into a more useful form. For the parabola, with the $\"x%5E2\"$ term, the form we want is:
\n" ); document.write( " $\"4p%28y-k%29+=+%28x-h%29%5E2\"$
\n" ); document.write( "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\".

\n" ); document.write( "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 $\"x%5E2\"$ ). 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:
\n" ); document.write( " $\"y+%2B+9+=+x%5E2+-+8x\"$
\n" ); document.write( "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:
\n" ); document.write( " $\"y+%2B+25+=+x%5E2+-+8x+%2B+16\"$
\n" ); document.write( "The right side is now a perfect square. It is $\"%28x+-+4%29%5E2\"$ . (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:
\n" ); document.write( " $\"y+-+%28-25%29+=+%28x-4%29%5E2\"$
\n" ); document.write( "Last of all we need something for the \"4p\" in the desired form. We can always factor out a 1!
\n" ); document.write( " $\"1%28y+-+%28-25%29%29+=+%28x-4%29%5E2\"$
\n" ); document.write( "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 $\"x%5E2\"$ ) so the focus is \"p\" units up from the vertex and the directrix is a horizontal line \"p\" units down from the vertex.

\n" ); document.write( "So the focus is (4, $\"-25+%2B+1%2F4\"$ ) or (4, $\"-24%263%2F4\"$ ) and the directrix is the line: y = $\"-25+-+1%2F4\"$ or y = $\"-25%261%2F4\"$ \n" ); document.write( "

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