Final answer:
The focus of the parabola y = 8x² + 12 is (0, 1/32), the directrix is y = -1/32, and the focal diameter is 1/8. The graph is an upward opening parabola with the vertex at the origin.
Step-by-step explanation:
The given equation is 8 x² + 12, set equal to y, which doesn't directly fit the standard form of a parabolic equation. However, if we assume the equation is y = 8x² + 12, we can proceed with the analysis of the parabola. The standard form of a parabola opening upwards or downwards is y = ax² + bx + c, where (h, k) is the vertex of the parabola.
To find the focus, we use the formula 4p = 1/a, where a is the coefficient of x². In this case, a is 8. The focal length p is 1/(4a), so p = 1/(4*8), and we get p = 1/32. The vertex is at the origin (0,0), so the focus, being p units above the vertex for a parabola that opens upwards, is at (0, 1/32).
The directrix is a horizontal line p units below the vertex, so its equation is y = -1/32.
The focal diameter is the length of the line segment that is perpendicular to the axis of symmetry and passes through the focus, which is 4p units long. So the focal diameter is 1/8.
A basic sketch of the parabola would show it opening upwards with the vertex at the origin, the focus just above the vertex, and a horizontal directrix just below the vertex. A vertical line through the focus would represent the axis of symmetry.