Problem 10 - College Algebra

Question

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

Step-by-Step Solution

Answer

The focus of the parabola is at the point (0,2), and the equation of the directrix is \(y=-2\). The graph is a parabola opening upwards, with vertex at the origin, focus at (0,2) and directrix at \(y=-2\).

Step 1: Identify the 'a' value.

Identify the value of \(a\) by comparing the given equation with the standard equation. In the standard equation \(x^{2}=4 a y\), \(4a=8\) implies \(a=2\).

Step 2: Determine the Focus.

The focus lies \(a\) units above the vertex on the axis of the parabola. For this equation, the vertex is at the origin \((0,0)\), so the focus is at \((0,2)\). The general coordinates for the focus of a parabola with vertex at the origin are \((0,a)\).

Step 3: Determine the Directrix.

The directrix is a horizontal line that is \(a\) units below the vertex along the axis of the parabola. So it follows the equation \(y=-a\). Here, the equation of the directrix is \(y=-2\).

Step 4: Graph the Parabola.

Draw the graph of the equation, marking the focus and the directrix. The vertex is at the origin, the focus is at the point (0,2), and the directrix is the horizontal line \(y=-2\). The parabola is open upwards, as dictated by the form of the equation.