Question 1) Vertex of the Parabola.
[tex]y-3=12(x+5)^{2} \\ \\
y= 12(x+5)^{2}+3[/tex]
The vertex of the general parabolic equation:
[tex]y=a(x-h)^{2}+k [/tex]
lies at (h,k)
Comparing the given equation to general equation, we can write:
h = - 5
k = 3
So, the vertex of the given parabola will be (-5, 3)
Question 2) Equation of Directrix
The correct equation of the parabola is:
[tex]y= \frac{1}{2} x^{2} +6x+23 \\ \\
[/tex]
First we need to convert the equation to standard form as shown below:
[tex]y= \frac{1}{2}( x^{2} +12x)+23 \\ \\
y= \frac{1}{2}( x^{2} +12x+36)+23- \frac{1}{2}(36) \\ \\
y= \frac{1}{2}(x+6)^{2}+5 \\ \\
y-5= \frac{1}{2}(x+6)^{2} \\ \\
2(y-5)=(x+6)^{2} \\ \\
4( \frac{1}{2})(y-5)= (x+6)^{2}[/tex]
The directrix of the general parabola of the form:
[tex]4p(y-k)=(x-h)^{2} [/tex]
is y = k - p
Comparing equation of given parabola with general parabolic equation, we can write:
p =1/2
k = 5
So, equation of directirx will be:
y = 5 - 1/2 = 4.5
So, option B gives the correct answer.
Question 3) Focus of the parabola
The correct equation of the parabola is:
[tex] \frac{1}{32}(y-2)^{2}=x-1 \\ \\
(y-2)^{2}=32(x-1) \\ \\
(y-2)^{2}=4*8(x-1)[/tex]
Comparing this equation to the general parabolic equation, we can write:
p=8
h =1
k = 2
The focus of the parabola will be at (h+p,k) = (9,2)
So the focus of the parabola is at (9,2)
Thus, option D gives the correct answer.
