For the parabola given by (y – 3)^2 = 8(x + 5), find the following:
(a) direction parabola opens (to the left, right, up, or down) Answer: ___________
(b) vertex Answer: ___________
(c) focus Answer: ___________
To find the requested information for the given parabola equation (y – 3)^2 = 8(x + 5), we can use the standard form of a parabola equation, which is (y – k)^2 = 4a(x – h), where (h, k) represents the vertex coordinates and a determines the direction of the parabola.
(a) To determine the direction in which the parabola opens (left, right, up, or down), we need to examine the sign of the coefficient a. In this case, we can see that a = 2, which is positive. A positive value of a indicates that the parabola opens upward.
Therefore, the answer to part (a) is: Upward.
(b) The vertex is given by the coordinates (h, k). Comparing the given equation to the standard form, we can see that (h, k) = (-5, 3). Therefore, the vertex of the parabola is (-5, 3).
Therefore, the answer to part (b) is: Vertex = (-5, 3).
(c) To determine the focus of the parabola, we need to find the value of p, which represents the distance from the vertex to the focus. In this case, p = a/4 = 2/4 = 1/2.
Since the parabola opens upward, the focus is located above the vertex along the axis of symmetry. The focus will be at a distance of p = 1/2 units above the vertex.
To find the focus, we add the distance p to the y-coordinate of the vertex. So, the y-coordinate of the focus becomes k + p = 3 + 1/2 = 7/2.
Therefore, the focus of the parabola is (-5, 7/2).
Therefore, the answer to part (c) is: Focus = (-5, 7/2).