Question

find the vertex, focus, and directrix of the parabola without completing the square, and determine whether the parabola opens upward or downward.

( y=\frac{1}{20} x^{2}-2 )

the coordinates of the vertex are ( (0,-2) ).
(simplify your answer. type an ordered pair.)

the coordinates of the focus are ( square ).
(simplify your answer. 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 equation \(y=\frac{1}{20}x^{2}-2\), we have \(h = 0\), \(k=-2\). The general formula for a parabola \(y=ax^{2}+bx + c\) (in our case \(b = 0\), \(c=-2\)) and for a parabola \(y=a(x - h)^2+k\), the relationship between \(a\) and the focus is given by the formula \(a=\frac{1}{4p}\), where \(p\) is the distance from the vertex to the focus.

Step2: Find the value of \(p\)

Given \(a=\frac{1}{20}\), from \(a=\frac{1}{4p}\), we can solve for \(p\). Cross - multiply: \(4p\times1=20\times1\), so \(p = 5\).

Step3: Determine the focus

Since the parabola is of the form \(y=a(x - h)^2 + k\) with \(a=\frac{1}{20}>0\) (opens upward), the focus of a parabola \(y=a(x - h)^2 + k\) is \((h,k + p)\). Substituting \(h = 0\), \(k=-2\), \(p = 5\), we get \((0,-2 + 5)\).

Answer:

\((0,3)\)