To find the vertex, focus, and directrix of the given parabola [tex]\((y - 4)^2 = 8(x - 2)\)[/tex], we need to rewrite it in the standard form of a parabola.
For a parabola in the form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], we can determine the vertex, focus, and directrix as follows:
1. Vertex: The vertex of the parabola is at the point [tex]\((h, k)\)[/tex].
2. Focus: The focus of the parabola is at the point [tex]\((h + p, k)\)[/tex].
3. Directrix: The directrix of the parabola is the line [tex]\(x = h - p\)[/tex].
Given the equation [tex]\((y - 4)^2 = 8(x - 2)\)[/tex]:
1. The standard form of the equation is [tex]\((y - k)^2 = 4p(x - h)\)[/tex].
- Compare [tex]\((y - 4)^2 = 8(x - 2)\)[/tex] with [tex]\((y - k)^2 = 4p(x - h)\)[/tex].
- Here, [tex]\(k = 4\)[/tex], [tex]\(h = 2\)[/tex], and [tex]\(4p = 8\)[/tex].
2. Solving for [tex]\(p\)[/tex]:
- [tex]\(4p = 8 \Rightarrow p = \frac{8}{4} = 2\)[/tex].
Now we have:
- Vertex: [tex]\((h, k) = (2, 4)\)[/tex]
- Focus: [tex]\((h + p, k) = (2 + 2, 4) = (4, 4)\)[/tex]
- Directrix: [tex]\(x = h - p = 2 - 2 = 0\)[/tex]
Thus, the correct answer is:
C. vertex: [tex]\((2, 4)\)[/tex], focus: [tex]\((4, 4)\)[/tex], directrix: [tex]\(x = 0\)[/tex].
