Problem 11 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$x^{2}=-16 y$$

Step-by-Step Solution

Answer

The focus of the given parabola is at the point (0,-4) and the equation of the directrix is \(y=4\). The parabola opens downwards and its graphical representation is symmetrical about the y-axis.

Step 1: Identifying the form of parabola and determining 'a'

Identify that the given equation, \(x^{2}=-16y\), represents a parabola. Compare it with the standard equation of the parabola, which is \(x^{2} = 4ay\). Here, 'a' can be determined by comparing the equations. In this case, we have \(4a = -16\), solving this gives \(a= -4\).

Step 2: Finding the focus

The focus of the parabola is given by the point (0,a). Therefore, in this case, our focus is at the point (0,-4).

Step 3: Finding the directrix

The directrix of the parabola is the line given by the equation \(y=-a\). Hence, in this case, the equation of the directrix is \(y=4\).

Step 4: Drawing the graph

To graph the parabola, first plot the focus at (0,-4) and draw a horizontal line at \(y=4\) for the directrix. The vertex of the parabola is at the origin (0,0) since it is not translated. It's clear that the parabola opens downward because 'a' is negative. Starting at the vertex, sketch the parabola such that it is symmetrical about the y-axis and gets closer to the directrix, but does not touch or cross it. The focus should be within the parabola.