Question

Find the focus, directrix, vertex, and axis of symmetry for the parabola 8(y - 1) = (x - 3)²

a) Focus: (3, 1), Directrix: y = -1, Vertex: (3, 1), Axis of Symmetry: x = 3
b) Focus: (3, 1), Directrix: y = 3, Vertex: (3, 1), Axis of Symmetry: x = -1
c) Focus: (3, -1), Directrix: y = 1, Vertex: (3, -1), Axis of Symmetry: x = 3
d) Focus: (3, -1), Directrix: y = -3, Vertex: (3, -1), Axis of Symmetry: x = -3

Answer 1

The focus is (3, 3), the directrix is y = -1, the vertex is (3, 1), and the axis of symmetry is x = 3.

Step-by-step explanation:

To find the focus, directrix, vertex, and axis of symmetry for the given parabola 8(y - 1) = (x - 3)², we can start by putting it in the standard form of a parabola, which is (x - h)² = 4p(y - k). In this form, the vertex is (h, k) and the axis of symmetry is the vertical line x = h.

Comparing the given equation to the standard form, we have (x - 3)² = 8(y - 1). So, h = 3 and k = 1. Therefore, the vertex is (3, 1) and the axis of symmetry is x = 3.

To find the focus and directrix, we need to find the value of p. In the standard form, p represents half the distance between the vertex and the focus/directrix. In this case, p = 2.

For a parabola facing upwards or downwards, the focus is located at (h, k + p) and the directrix is given by the equation y = k - p. Plugging in the values we found, the focus is (3, 1 + 2) = (3, 3) and the directrix is y = 1 - 2 = -1.