Problem 37 - College Algebra

Question

Find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. $$(x+1)^{2}=-8(y+1)$$

Step-by-Step Solution

Answer

The vertex is at (-1, -1), the focus is at (-1, -3), and the equation for the directrix is y = 1.

Step 1: Identify the Constants

The given equation is \((x+1)^{2} = -8(y+1)\). Comparing this with \((x-h)^{2} = 4a(y-k)\), we get 'h' as -1, 'k' as -1, and 'a' as '-2'. Hence, the parabola opens downward.

Step 2: Find the Vertex

The vertex of the parabola will be at (h, k). Substituting the values of h and k that we identified, the vertex will be at (-1, -1).

Step 3: Find the Focus

The focus of the parabola will be at (h, k+a). Substituting the values of h, k and a that were identified, the focus will be at (-1, -1-2) = (-1, -3).

Step 4: Find the Directrix

The equation of the directrix will be y = k - a. Substituting the values of k and a that we identified, the equation of the directrix will be y = -1-(-2) = y = 1.

Step 5: Graph the Parabola

Finally, using the vertex, focus, and directrix, we will be able to sketch the graph of the parabola. Remember that the parabola is a mirror image about its directrix, and is symmetrical around the vertex.