To analyze the parabola given by the equation x = 4y², we first reformat it to identify its key characteristics. This form indicates that we have a horizontally oriented parabola.
Vertex: The vertex of this parabola is at the origin, which is the point (0, 0).
Focus: For a parabola, the distance from the vertex to the focus is given by the formula p = 1/4a, where a is the coefficient of y² in the standard form x = 4py. Here, a = 4, so we have p = 1. The focus is located 1 unit to the right of the vertex, which gives us the focus at the point (1, 0).
Directrix: The directrix of the parabola can be found by moving p units to the left of the vertex. Since p = 1, the equation of the directrix is x = -1.
Focal Width: The focal width of the parabola, also known as the latus rectum, is the distance across the parabola through the focus. The length of the focal width is given by 4p, which means in our case it would be 4 × 1 = 4.
To summarize:
- Vertex: (0, 0)
- Focus: (1, 0)
- Directrix: x = -1
- Focal Width: 4