Problem 15 - College Algebra

Question

Find the focus and directrix of the parabola with the given equation. Then graph the parabola. $$8 x^{2}+4 y=0$$

Step-by-Step Solution

Answer

The focus of the parabola is at (0,-0.5) and the directrix is the line y = 0.5.

Step 1: Rearrange the parabola equation.

Rearrange the given equation to look like the standard form. The given equation is \(8x^{2} + 4y = 0\). Divide every term by 4 to simplify it and the equation becomes \(2x^{2} + y = 0\). Rewrite this equation in the standard form, it results in \(y = -2x^{2}\), which resembles to the standard form \(4p(y-k) = (x-h)^{2}\). Here, the vertex (h,k) is (0,0).

Step 2: Find the value of 'p'.

Refer to the standard form, the number in front of y is -2 which equals to \(4p\), so \(p = -2/4 = -0.5\).

Step 3: Calculate the focus and directrix.

Use the formula for the focus of a parabola that opens downward: (h, k+p), we have the focus as (0, -0.5). As for the directrix, it's the horizontal line k - p, so y = 0.5.

Step 4: Graph the parabola, focus and directrix.

To graph the parabola, first, plot the vertex which is at the origin (0,0), downward focus (0, -0.5) and the directrix line at y = 0.5. Next, draw the parabolic curve such that it passes through the vertex and it is equidistant to the focus and directrix.