Problem 38 - College Algebra

Question

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(x+2)^{2}=-8(y+2)$$

Step-by-Step Solution

Answer

The vertex of the parabola is (-2,-2), the focus is (-2, 0) and the directrix is y=-4.

Step 1: Identify the Vertex

From the formula, the coordinates for the vertex (h, k) pairs with the values being subtracted in the equation, that is (-2,-2).

Step 2: Calculate the value of p

In the given equation \((x+2)^{2}=-8(y+2)\), comparing it with the standard form \(4p = -8\), we divide -8 by 4 to find the value of p. Thus, p = -8/4 = -2.

Step 3: Find Focus and Directrix

For a parabola opening downward, we use the formula for the focus and directrix with k-p and k+p respectively. The focus will be the point (-2, -2 - -2) = (-2, 0) and the directrix will be y = -2 - 2 = -4.

Step 4: Graph the parabola

Plot the vertex point (-2,-2) along with the focus (-2, 0). Draw the directrix line at y = -4. Sketch the parabola opening downward to pass through the focus and mirror against the directrix.