Problem 12 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$x^{2}=-20 y$$

Step-by-Step Solution

Answer

The focus of the parabola is at (0, -5) and the directrix is \(y = 5\).

Step 1: Identifying the value of p

The given equation is \(x^2 = -20y\). Comparing this with the standard form \(x^2 = 4py\), we can see that \(4p = -20\). Solving this for \(p\), we get \(p = -20 / 4 = -5\).

Step 2: Finding the focus of the parabola

The focus of a parabola given in the form \(x^2 = 4py\) is at the coordinates (0,p). Since the value of \(p\) in this case is -5, the focus will be (0, -5).

Step 3: Finding the directrix of the parabola

The equation of the directrix for a parabola in the form \(x^2 = 4py\) is \(y = -p\). Therefore, because \(p = -5\), the equation of the directrix is \(y = 5\).

Step 4: Graphing the Parabola

To graph the parabola, plot the vertex at (0,0). Then plot the focus at (0,-5), and draw the directrix at \(y = 5\). The parabola opens downward because \(p\) is negative, and its width is determined by the value of \(p\). What you will then have is a parabola with a vertex at the origin (0,0), focus at (0,-5) and directrix at \(y = 5\).