Problem 35 - College Algebra

Question

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

Step-by-Step Solution

Answer

The vertex of the parabola is (2, 1), the focus is (2, 3), and the directrix is y = -1.

Step 1: Identifying the Vertex

The vertex is given by the (h, k) in the standard form of the parabola. In the equation \((x-2)^{2}=8(y-1)\), comparing with the standard form, we see that h = 2 and k = 1. Therefore, the vertex of the parabola is at the point (2, 1).

Step 2: Finding the value of p

In the equation \((x-2)^{2}=8(y-1)\), the coefficient on the right is 8, which is 4p. Solving the equation 4p = 8 gives p = 2.

Step 3: Determining the Focus and Directrix

Since the equation is in the form \((x-h)^2 = 4p(y-k)\), and we have determined that p = 2, we know that the parabola opens upward. Therefore, the focus is (h, k + p) = (2, 1 + 2) = (2, 3), and the equation of the directrix is y = k - p = 1 - 2 = -1.

Step 4: Graphing the Parabola

To graph this parabola, start by plotting the vertex at the point (2, 1). Then plot the focus at (2, 3) and draw the line y = -1 for the directrix. Sketch the parabola so it opens upward, with the vertex at the minimum point, passing through the focus, and getting closer to the directrix as the curve extends in both directions.