Question

find the focus and directrix of the parabola with the given equation. then graph the parabola.
$x^2 = -24y$

the focus is \boxed{}.
(type an ordered pair.)

the directrix is \boxed{}.
(type an equation.)

use the graphing tool to graph the parabola only.
click to enlarge graph

Explanation:

Step1: Recall parabola standard form

The standard form of a parabola that opens up or down is \(x^{2}=4py\), where the focus is \((0,p)\) and the directrix is \(y = -p\). If \(p>0\), it opens up; if \(p<0\), it opens down.
Given equation: \(x^{2}=-24y\). Compare with \(x^{2}=4py\), so \(4p=-24\).

Step2: Solve for \(p\)

Divide both sides of \(4p=-24\) by 4: \(p=\frac{-24}{4}=-6\).

Step3: Find focus

Using the standard form, focus is \((0,p)\). Substitute \(p = -6\), so focus is \((0,-6)\).

Step4: Find directrix

Directrix is \(y=-p\). Substitute \(p=-6\), so \(-p = 6\), thus directrix is \(y = 6\).

Answer:

The focus is \((0, -6)\).
The directrix is \(y = 6\).