Question

find the vertex, focus, and directrix of the parabola without completing the square, and determine whether the parabola open

( 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 ) ( = square ).
(simplify your answer.)

Explanation:

Step1: Recall the standard form of parabola equation

The standard form of a parabola that opens up or down is \(x^{2}=4p(y - k)\), where \((k,p + k)\) is the focus and \(y=k - p\) is the directrix, and \((k,k)\) is the vertex.
The given equation \(y=\frac{1}{20}x^{2}-2\) can be rewritten as \(x^{2}=20(y + 2)\).
Comparing with \(x^{2}=4p(y - k)\), we have \(4p = 20\), so \(p=5\).

Step2: Find the directrix

Since \(k=-2\) (from the vertex \((0,-2)\)) and \(p = 5\), using the formula \(y=k - p\).
Substitute \(k=-2\) and \(p = 5\) into \(y=k - p\), we get \(y=-2-5\).

Answer:

\(y=-7\)