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 (0,3).
(simplify your answer. type an ordered pair.)
the directrix is y = - 7.
(simplify your answer.)
does the parabola open upward or downward?
upward
downward

Explanation:

Step1: Recall the standard form of parabola

The standard form of a parabola is \(x^{2}=4py\). For the given equation \(y = \frac{1}{20}x^{2}-2\), we can rewrite it as \(x^{2}=20(y + 2)\).

Step2: Identify the value of \(p\)

Comparing \(x^{2}=20(y + 2)\) with \(x^{2}=4py\), we get \(4p = 20\), so \(p=5\).

Step3: Determine the direction of parabola

Since the coefficient of \(x^{2}\) in \(y=\frac{1}{20}x^{2}-2\) is positive (\(\frac{1}{20}>0\)), the parabola opens upward.

Answer:

Upward