Identify the vertex, focus and directrix of -x^2 - 8x + y - 18 = 0
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Identify the vertex, focus and directrix of -x^2 - 8x + y - 18 = 0

[From: ] [author: ] [Date: 12-05-23] [Hit: ]
A parabola in this form has its vertex at (h, k),Now, we need to put the given equation into this form. To do this, notice that this form has all terms involving y on one side and all terms involving x on the other.......
-x^2 - 8x + y - 18 = 0

Please Help, I don't understand problems like this, Please show steps.

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Notice that we have two variants for the standard equation of a parabola:
y - k = [1/(4p)](x - h)^2 OR x - h = [1/(4p)](y - k)^2.

In order to make our solution, notice that the first flavor has a x^2 term, while the second has a y^2 term. The equation given has a x^2 term, so we need to use:
y - k = [1/(4p)](x - h)^2.

A parabola in this form has its vertex at (h, k), its focus |p| units directly above the vertex for p > 0 (if p < 0, then the focus is below the vertex instead), and its directix is a horizontal line |p| units below the vertex for p > 0 (if p < 0, then the directix lies above the vertex instead).

Now, we need to put the given equation into this form. To do this, notice that this form has all terms involving y on one side and all terms involving x on the other. Doing this with the given equation yields:
y - 18 = x^2 + 8x.

At this point, we need to complete the square on the right side. Recall that in order to complete the square, we need to take the x coefficient, halve it, and then square the result. Here, the x coefficient is 8, which halves to get 4, and squares to get 16. So, adding 16 to both sides gives:
y - 18 + 16 = x^2 + 8x + 16 ==> y - 2 = (x + 4)^2.

Note that the right side is a perfect square binomial and factored as such.

Now, the 1/(4p) term is just the number outside on the right side. There is an understood 1 on the right side since:
(x + 4)^2 = 1(x + 4)^2.

Then, we see that:
1/(4p) = 1 ==> p = 1/4.

So, the given equation in standard form is:
y - 2 = [1/)4 * 1/4)](x + 4)^2.

Comparing this to the standard equation yields the vertex to be (-4, 2). Then, since p = 1/4, the focus lies 1/4 units directly above the vertex at (-4, 2 + 1/4) = (-4, 9/4), and the directix is a horizontal line 1/4 units below the vertex---this has equation y = 2 - 1/4 = 7/4.

I hope this helps!
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