To analyze the parabola given by the equation x² = 12y, we need to identify several key components: the vertex, focus, directrix, and focal width.
Firstly, we can rewrite the equation in the standard form of a parabola that opens upwards, which is (x – h)² = 4p(y – k). In this case, our equation x² = 12y can be compared to (x – 0)² = 4p(y – 0). From this, we see that:
- The vertex (h, k) is at the point (0, 0).
- From the equation, we can deduce that 4p = 12, hence p = 3.
Now, the focus of the parabola is located at the point (h, k + p), which in our case is:
- Focus: (0, 0 + 3) = (0, 3).
The directrix, on the other hand, is a line located at y = k – p. So, for our parabola:
- Directrix: y = 0 – 3 = -3.
Lastly, the focal width of a parabola is given by the absolute value of 4p. Since we found 4p = 12, the focal width of our parabola is:
- Focal Width: 12.
In conclusion, for the parabola represented by the equation x² = 12y, we summarize:
- Vertex: (0, 0)
- Focus: (0, 3)
- Directrix: y = -3
- Focal Width: 12