Sure, let's go through the key features of the parabola given by the equation [tex]\( y^2 = 8x \)[/tex].
1. Vertex:
- The equation is in the form [tex]\( y^2 = 4px \)[/tex], where [tex]\( p \)[/tex] is a constant.
- For parabolas of the form [tex]\( y^2 = 4px \)[/tex], the vertex is located at the origin, [tex]\((0,0)\)[/tex].
Therefore, the vertex of the parabola [tex]\( y^2 = 8x \)[/tex] is at [tex]\((0, 0)\)[/tex].
2. Axis of Symmetry:
- For parabolas of the form [tex]\( y^2 = 4px \)[/tex], the axis of symmetry is the line along which the parabola is symmetrical.
- In this case, the parabola is symmetrical along the x-axis.
Thus, the axis of symmetry is the x-axis.
3. Focus:
- For parabolas of the form [tex]\( y^2 = 4px \)[/tex], the focus is located at [tex]\((p, 0)\)[/tex].
- Here, by comparing [tex]\( y^2 = 8x \)[/tex] with [tex]\( y^2 = 4px \)[/tex], we get [tex]\( 4p = 8 \)[/tex] which implies [tex]\( p = 2 \)[/tex].
- Therefore, the focus of the parabola [tex]\( y^2 = 8x \)[/tex] is at the point [tex]\((2, 0)\)[/tex].
So, the focus is at [tex]\((2, 0)\)[/tex].
4. Directrix:
- For parabolas of the form [tex]\( y^2 = 4px \)[/tex], the directrix is a line given by [tex]\( x = -p \)[/tex].
- Given [tex]\( p = 2 \)[/tex] from earlier, the directrix is the line [tex]\( x = -2 \)[/tex].
Thus, the directrix of the parabola is [tex]\( x = -2 \)[/tex].
5. Direction of Opening:
- For the parabola [tex]\( y^2 = 8x \)[/tex], the coefficient of [tex]\( x \)[/tex] is positive.
- For parabolas in the form [tex]\( y^2 = 4px \)[/tex] where [tex]\( p > 0 \)[/tex], they open to the right.
Therefore, the direction of opening is to the right.
Summarizing all the key features:
- The vertex is [tex]\((0, 0)\)[/tex].
- The axis of symmetry is the x-axis.
- The focus is at [tex]\((2, 0)\)[/tex].
- The directrix is [tex]\( x = -2 \)[/tex].
- The parabola opens to the right.
