Problem 7 - College Algebra

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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 \(a = 2\).

Step 3: 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 \(x = 2\).

Step 4: 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.