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
(type an ordered pair.)

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 \((h,k+\frac{1}{4a})\). Here \(a=\frac{1}{4}\).

Step2: Calculate the \(y\) - coordinate of the focus

Substitute \(a=\frac{1}{4}\), \(h=-4\), \(k = 0\) into the formula for the focus. First, find \(\frac{1}{4a}\). Since \(a=\frac{1}{4}\), then \(\frac{1}{4a}=\frac{1}{4\times\frac{1}{4}} = 1\).
The \(y\) - coordinate of the focus is \(k+\frac{1}{4a}=0 + 1=1\), and the \(x\) - coordinate is \(h=-4\).

Answer:

\((-4,1)\)