Let's analyze the equation of the parabola given by [tex]\( x = \frac{1}{12} y^2 \)[/tex]:
### A. Direction in Which the Parabola Opens
The equation is in the form [tex]\( x = a y^2 \)[/tex], where [tex]\( a \)[/tex] is a positive constant ([tex]\( \frac{1}{12} \)[/tex]). For parabolas in this form, the presence of the [tex]\( y^2 \)[/tex] term (and the absence of the [tex]\( x^2 \)[/tex] term) indicates that the parabola opens either to the right or to the left.
Since [tex]\( a \)[/tex] is positive, the parabola opens to the right.
### B. The Vertex of the Parabola
The general form of the equation is given as [tex]\( x = a (y - k)^2 + h \)[/tex]. In our equation, there are no [tex]\( h \)[/tex] and [tex]\( k \)[/tex] terms present. This implies that the vertex is at the origin.
Thus, the vertex is at [tex]\((0, 0)\)[/tex].
### C. The Location of the Focus
For a parabola of the form [tex]\( x = a y^2 \)[/tex], the focus is located at [tex]\((p + h, k)\)[/tex], where [tex]\( p = \frac{1}{4a} \)[/tex].
Given [tex]\( a = \frac{1}{12} \)[/tex]:
[tex]\[
p = \frac{1}{4 \cdot \frac{1}{12}} = \frac{1}{\frac{1}{3}} = 3
\][/tex]
Since [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are 0 (from the vertex coordinates), the focus is at:
[tex]\[
(p, 0) = (3, 0)
\][/tex]
### D. The Equation of the Directrix
The directrix of the parabola is given by the line [tex]\( x = h - p \)[/tex].
Given [tex]\( h = 0 \)[/tex] and [tex]\( p = 3 \)[/tex]:
[tex]\[
x = 0 - 3 = -3
\][/tex]
Therefore, the equation of the directrix is [tex]\( x = -3 \)[/tex].
### E. The Equation of the Axis of Symmetry
For parabolas that open to the right or left, the axis of symmetry is a vertical line passing through the vertex. Since our vertex is [tex]\((0, 0)\)[/tex]:
The axis of symmetry is along the y-axis, so its equation is:
[tex]\[
y = 0
\][/tex]
### Summary
Thus, for the equation [tex]\( x = \frac{1}{12} y^2 \)[/tex]:
A. The parabola opens to the right.
B. The vertex is at [tex]\((0, 0)\)[/tex].
C. The focus is at [tex]\((3.0, 0)\)[/tex].
D. The equation of the directrix is [tex]\( x = -3.0 \)[/tex].
E. The equation of the axis of symmetry is [tex]\( y = 0 \)[/tex].
