Problem 16 - College Algebra

Question

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

Step-by-Step Solution

Answer

The focus of the given parabola is at (-1/8, 0) and the equation of the directrix is x=1/8. This is a graph of a parabola that opens to the left, with its vertex at the origin.

Step 1: Determining the Parabola's Orientation and Calculating the Distance

The equation of the parabola is \(8y^2 + 4x = 0\). Here, \(A=8\) and \(B=4\). So, because B is positive, the parabola opens to the left. The distance to the focus/directrix from the vertex (p) is calculated using the formula \(-B/4A\). Substituting values, we get \(p = -4/(4*8) = -1/8\).

Step 2: Finding the Focus and the Directrix

For parabolas that open left, the focus is at \((h - p, k)\) and the directrix is at \(x = h + p\), where \((h, k)\) are the coordinates of the parabola's vertex. Here, as the given equation is an translation of the standard parabolic equation \(y^2 = 4px\), our vertex at the origin \((h, k) = (0, 0)\). Substituting these and the calculated p value into the formulas, we get the focus at \((-1/8, 0)\) and the directrix at \(x= 1/8\).

Step 3: Graphing the Parabola

Plot the vertex, focus and the directrix line on the graph. The vertex is at the origin \((0,0)\). Plot point at the focus \((-1/8, 0)\). Draw a vertical line at \(x=1/8\) to represent the directrix. Draw curves from the vertex such that they bend towards the focus and are equidistant from the directrix.