Question

what are the vertex, focus, and directrix of the parabola with the equation $x^2 + 8x + 4y + 4 = 0$?
vertex $(-4, 3)$; focus $(-4, 2)$; directrix $y = 4$
vertex $(4, -3)$; focus $(4, -7)$; directrix $y = 1$
vertex $(4, -3)$; focus $(4, -4)$; directrix $y = -2$
vertex $(-4, 3)$; focus $(-4, -1)$; directrix $y = 7$

Explanation:

Step1: Rewrite the equation

Start with the given equation \(x^{2}+8x + 4y+4 = 0\). Isolate the \(x\)-terms: \(x^{2}+8x=-4y - 4\).
Complete the square for the \(x\)-terms. The coefficient of \(x\) is 8, half of it is 4, and squaring it gives 16. Add 16 to both sides:
\(x^{2}+8x + 16=-4y-4 + 16\)
Which simplifies to \((x + 4)^{2}=-4y+12\), or \((x + 4)^{2}=-4(y - 3)\).

Step2: Identify the vertex

The standard form of a parabola that opens up or down is \((x - h)^{2}=4p(y - k)\), where \((h,k)\) is the vertex. Comparing \((x + 4)^{2}=-4(y - 3)\) with the standard form, we have \(h=-4\), \(k = 3\), so the vertex is \((-4,3)\).

Step3: Find \(p\) and determine focus/directrix

From \((x + 4)^{2}=-4(y - 3)\), we have \(4p=-4\), so \(p=-1\). Since \(p\) is negative, the parabola opens downward.
The focus is at \((h,k + p)\), so substituting \(h=-4\), \(k = 3\), \(p=-1\), we get the focus as \((-4,3-1)=(-4,2)\).
The directrix is the line \(y=k - p\), so \(y=3-(-1)=4\)? Wait, no: wait, the formula for directrix when the parabola is \((x - h)^{2}=4p(y - k)\) is \(y=k - p\) when \(p\) is the value from the equation. Wait, in our equation, \(4p=-4\), so \(p=-1\). So the directrix is \(y=k - p\)? Wait, no, let's recall: for \((x - h)^{2}=4p(y - k)\), if \(p>0\) opens up, focus \((h,k + p)\), directrix \(y=k - p\); if \(p<0\) opens down, focus \((h,k + p)\), directrix \(y=k - p\) (since \(p\) is negative, \(k - p\) is \(k+|p|\)). Wait, in our case, \(4p=-4\), so \(p=-1\). Then focus is \((h,k + p)=(-4,3+(-1))=(-4,2)\). Directrix is \(y=k - p=3-(-1)=4\)? Wait, no, wait: the standard form is \((x - h)^2 = 4p(y - k)\), so the directrix is \(y = k - p\). Wait, if \(p=-1\), then \(y=3-(-1)=4\)? Wait, but let's check the options. The first option has vertex \((-4,3)\), focus \((-4,2)\), directrix \(y = 4\), which matches our calculations.

Answer:

vertex \((-4, 3)\); focus \((-4, 2)\); directrix \(y = 4\) (the first option)