Question

determine the vertex, focus, and directrix for the parabola.

( y=\frac{1}{4}(x + 4)^{2} )

the vertex is ( (-4,0) ).
(type an ordered pair.)

the focus is ( (-4,1) ).
(type an ordered pair.)

the directrix is ( y=square ).

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola is \(y=a(x - h)^2+k\), where \((h,k)\) is the vertex. For the given parabola \(y=\frac{1}{4}(x + 4)^2\), we have \(h=-4\), \(k = 0\). The formula for the focus of a parabola \(y=a(x - h)^2+k\) is \((h,k+\frac{1}{4a})\), and the formula for the directrix is \(y=k-\frac{1}{4a}\).

Step2: Find the value of \(a\)

From \(y=\frac{1}{4}(x + 4)^2\), we can see that \(a=\frac{1}{4}\).

Step3: Calculate the directrix

Substitute \(k = 0\) and \(a=\frac{1}{4}\) into the formula \(y=k-\frac{1}{4a}\).

$$
LATEXBLOCK0
$$

Answer:

\(y=-1\)