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Chapter 7: Problem 7

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

Short Answer

Expert verified
The focus of the parabola is at \((-2,0)\) and the directrix is at \(x=2\). The parabola opens to the left with the vertex at the origin.

Step by step solution

01

Identify the Form of the Equation

Identify and compare the given equation \(y^{2} = -8x\) with the standard equation of a parabola. As it matches with the form \(y^{2} = -4ax\), the parabola opens to the left.
02

Calculate the Value of the Parameter a

The parameter \(a\) in the standard equation represents the distance from the vertex to the focus and vertex to the directrix. From the given equation \(y^{2} = -8x\), equate \(-4a\) to \(-8\) to get the value of \(a\). This gives \(a = 2\).
03

Determine the Focus and Directrix

Using the value of \(a\), calculate the focus and directrix. The focus, given by \((-a,0)\), will be \((-2,0)\), and the directrix, given by \(x = a\), will be \(x = 2\).
04

Graph the Parabola

Draw the parabola by first marking the focus and directrix. Since the parabola opens to the left, draw a symmetric curve with the focus inside and the curve getting increasingly distant from the directrix line as it extends.

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Most popular questions from this chapter

The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 4 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((0,15) ;\) Directrix: \(y=-15\)

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((9,0) ;\) Directrix: \(x=-9\)

Find the standard form of the equation of each parabola satisfying the given conditions. Focus: \((0,20) ;\) Directrix: \(y=-20\)
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