Let’s determine the equation of the directrix for a parabola given the vertex and the focus.
1. Identify the given data: The vertex of the parabola is at the origin [tex]\((0,0)\)[/tex] and the focus is at [tex]\((-2,0)\)[/tex].
2. Understand the relationship between the vertex, focus, and directrix:
- For a parabola with a horizontal axis of symmetry (opening left or right), the directrix is a vertical line.
- The distance from the vertex to the focus is equal to the distance from the vertex to the directrix.
- The focus is [tex]\((-2,0)\)[/tex], which is 2 units to the left of the vertex.
3. Calculate the location of the directrix: Since the focus is 2 units to the left of the vertex, the directrix will be 2 units to the right of the vertex.
- The directrix is equidistant on the opposite side from the vertex.
- Therefore, since the vertex is at [tex]\( (0,0) \)[/tex], moving 2 units to the right of the vertex, the directrix will be at [tex]\( x = 2 \)[/tex].
4. Write the equation of the directrix: The equation for a vertical line 2 units to the right of the vertex ([tex]\(0,0\)[/tex]) is:
[tex]\[
x = 2
\][/tex]
Thus, the correct equation for the directrix of the parabola is [tex]\( \boxed{x = 2} \)[/tex].
