To determine the direction in which the parabola opens, we need to understand the relationship between the vertex, directrix, and focus of the parabola.
Since we are given the vertex at [tex]\((0,0)\)[/tex] and the directrix as the vertical line [tex]\(x = -4\)[/tex]:
1. Locate the directrix: The directrix is a vertical line located at [tex]\(x = -4\)[/tex]. This means it is positioned 4 units to the left of the y-axis.
2. Understand the properties of the parabola:
- The vertex of the parabola is the point [tex]\((0,0)\)[/tex].
- The parabola will open towards the focus and away from the directrix.
- The focus of the parabola will be an equal distance from the vertex but on the opposite side of the directrix.
3. Determine the focus:
- The vertex is at [tex]\( (0,0) \)[/tex].
- Given the directrix is [tex]\( x = -4 \)[/tex], the distance from the directrix to the vertex is [tex]\(4\)[/tex] units.
- The focus must be on the opposite side of the vertex from the directrix, thus it must be 4 units to the right of the y-axis.
- Therefore, the focus is at [tex]\( (4,0) \)[/tex].
4. Direction of opening:
- The parabola opens towards the focus.
- Since the focus is to the right of the vertex at [tex]\((4,0)\)[/tex], the parabola opens to the right.
Thus, the parabola opens to the right.
