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.

a. [tex]x=\frac{1}{12} y^2[/tex]

b. [tex]y-2=-\frac{1}{16}(x-1)^2[/tex]



Answer :

Sure! Let’s go through each equation step-by-step to determine each of the requested properties of the parabolas.

##################
### Equation a ###
##################

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

A. If the parabola opens up, down, left, or right:

The given equation is in the form [tex]\( x = a(y - k)^2 + h \)[/tex]. Here, [tex]\( a = \frac{1}{12} \)[/tex], and since [tex]\( a \)[/tex] is positive, the parabola opens to the right.

B. The vertex of the parabola:

For the equation [tex]\( x = \frac{1}{12} y^2 \)[/tex], the vertex [tex]\( (h, k) \)[/tex] is [tex]\((0, 0)\)[/tex].

C. The location of the focus:

The focus of a parabola [tex]\( x = a(y - k)^2 + h \)[/tex] is located at [tex]\( \left(h + \frac{1}{4a}, k\right) \)[/tex]. Here [tex]\( a = \frac{1}{12} \)[/tex], so the focus is:
[tex]\[ \left(0 + \frac{1}{4 \times \frac{1}{12}}, 0\right) = \left(3, 0\right) \][/tex]

D. The equation of the directrix:

The directrix of a parabola [tex]\( x = a(y - k)^2 + h \)[/tex] has the equation [tex]\( x = h - \frac{1}{4a} \)[/tex]. Here [tex]\( a = \frac{1}{12} \)[/tex], so the equation of the directrix is:
[tex]\[ x = 0 - \frac{1}{4 \times \frac{1}{12}} = -3 \][/tex]

E. The equation of the axis of symmetry:

The axis of symmetry for the parabola [tex]\( x = a(y - k)^2 + h \)[/tex] is the line [tex]\( y = k \)[/tex]. Here [tex]\( k = 0 \)[/tex], so the equation of the axis of symmetry is:
[tex]\[ y = 0 \][/tex]

##################
### Equation b ###
##################

### Given equation:
[tex]\[ y - 2 = -\frac{1}{16} (x - 1)^2 \][/tex]

A. If the parabola opens up, down, left, or right:

The given equation is in the form [tex]\( y - k = a(x - h)^2 \)[/tex]. Here, [tex]\( a = -\frac{1}{16} \)[/tex], and since [tex]\( a \)[/tex] is negative, the parabola opens downwards.

B. The vertex of the parabola:

For the equation [tex]\( y - 2 = -\frac{1}{16}(x - 1)^2 \)[/tex], the vertex [tex]\( (h, k) \)[/tex] is [tex]\((1, 2)\)[/tex].

C. The location of the focus:

The focus of a parabola [tex]\( y - k = a(x - h)^2 \)[/tex] is located at [tex]\( \left(h, k + \frac{1}{4a}\right) \)[/tex]. Here [tex]\( a = -\frac{1}{16} \)[/tex], so the focus is:
[tex]\[ \left(1, 2 + \frac{1}{4 \times -\frac{1}{16}} \right) = \left(1, 6\right) \][/tex]

D. The equation of the directrix:

The directrix of a parabola [tex]\( y - k = a(x - h)^2 \)[/tex] has the equation [tex]\( y = k - \frac{1}{4a} \)[/tex]. Here [tex]\( a = -\frac{1}{16} \)[/tex], so the equation of the directrix is:
[tex]\[ y = 2 - \frac{1}{4 \times -\frac{1}{16}} = -2 \][/tex]

E. The equation of the axis of symmetry:

The axis of symmetry for the parabola [tex]\( y - k = a(x - h)^2 \)[/tex] is the line [tex]\( x = h \)[/tex]. Here [tex]\( h = 1 \)[/tex], so the equation of the axis of symmetry is:
[tex]\[ x = 1 \][/tex]

### Summary:

For equation [tex]\( a \)[/tex]:
A. Opens right
B. Vertex at [tex]\((0, 0)\)[/tex]
C. Focus at [tex]\((3, 0)\)[/tex]
D. Directrix: [tex]\(x = -3\)[/tex]
E. Axis of symmetry: [tex]\(y = 0\)[/tex]

For equation [tex]\( b \)[/tex]:
A. Opens down
B. Vertex at [tex]\((1, 2)\)[/tex]
C. Focus at [tex]\((1, 6)\)[/tex]
D. Directrix: [tex]\(y = -2\)[/tex]
E. Axis of symmetry: [tex]\(x = 1\)[/tex]