Final answer:
The parabola 2x^2 - 8y = 0 has its vertex at the origin (0,0), with the axis of symmetry along the y-axis. The focus is at (0, 1) and the directrix is the horizontal line y = -1.
Step-by-step explanation:
The equation of the parabola given is 2x^2 - 8y = 0. To find the axis, directrix, and focus, we first need to rewrite the equation in standard form. By dividing the entire equation by 2, we get x^2 = 4y, which is a parabola that opens upwards. The standard form of a parabola that opens vertically (upwards or downwards) is (x - h)^2 = 4p(y - k), where (h, k) is the vertex and p is the distance from the vertex to the focus and to the directrix.
In our equation, h = 0 and k = 0, since there is no (x - h) or (y - k) part, which means the vertex is at the origin (0, 0). The coefficient 4p in our equation is 4, so p = 1. The axis of symmetry is the y-axis, since the parabola opens upwards. The focus of the parabola is (0, p), so it's at (0, 1). The directrix is a horizontal line at y = -p, which means it's the line y = -1.