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For problems 7-8, determine the following for each equation:

A. If the parabola opens up, down, left, or right.
B. The vertex of the parabola.
C. The location of the focus.
D. The equation of the directrix.
E. The equation of the axis of symmetry.

Given: [tex]x=\frac{1}{12} y^2[/tex]

a. Determine if the parabola opens up, down, left, or right.
b. Find the vertex of the parabola.
c. Locate the focus of the parabola.
d. Write the equation of the directrix.
e. Write the equation of the axis of symmetry.

Sagot :

GinnyAnswer
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].
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