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Chapter 7: Problem 16

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

Short Answer

Expert verified
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 by step solution

01

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 \(p = -4/(4*8) = -1/8\).
02

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 \(x= 1/8\).
03

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.

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