An equation of a parabola is given. Write the equation of the parabola in standard form, and identify the
vertex and the focal diameter.
-$$y ^ { 2 } - 8 y - 12 x - 8 = 0$$
A) $( y - 4 ) ^ { 2 } = 12 ( x + 2 )$
vertex: $( - 2,4 )$; focal diameter: 12

B) $( y + 4 ) ^ { 2 } = 12 ( x - 2 )$
vertex: $( - 4,2 )$; focal diameter: 6
C) $( y - 4 ) ^ { 2 } = 12 ( x + 2 )$
vertex: $( - 2,4 )$; focal diameter: 6

D) $( y + 4 ) ^ { 2 } = 12 ( x - 2 )$
vertex: $( - 4,2 )$; focal diameter: 12

Question

An equation of a parabola is given. Write the equation of the parabola in standard form, and identify the
vertex and the focal diameter.
-$$y ^ { 2 } - 8 y - 12 x - 8 = 0$$
A) $( y - 4 ) ^ { 2 } = 12 ( x + 2 )$
vertex: $( - 2,4 )$; focal diameter: 12

B) $( y + 4 ) ^ { 2 } = 12 ( x - 2 )$
vertex: $( - 4,2 )$; focal diameter: 6
C) $( y - 4 ) ^ { 2 } = 12 ( x + 2 )$
vertex: $( - 2,4 )$; focal diameter: 6

D) $( y + 4 ) ^ { 2 } = 12 ( x - 2 )$
vertex: $( - 4,2 )$; focal diameter: 12

Quizplus · Accepted Answer

To convert the given equation $y^2 - 8y - 12x - 8 = 0$ to standard form, complete the square for the $y$ terms. First, isolate the $y$ terms and add $16$ (which is $(\frac{8}{2})^2$) to both sides to complete the square, resulting in $y^2 - 8y + 16 = 12x + 24$. This simplifies to $(y - 4)^2 = 12(x + 2)$, matching option A. The vertex form of a parabola is $(y - k)^2 = 4p(x - h)$, where $(h, k)$ is the vertex and $4p$ is the focal diameter. Thus, the vertex is $(-2, 4)$ and the focal diameter is $12$, confirming option A as correct.

An Equation of a Parabola Is Given y2−8y−12x−8=0y ^ { 2 } - 8 y - 12 x - 8 = 0y2−8y−12x−8=0

An Equation of a Parabola Is Given $y ^ { 2 } - 8 y - 12 x - 8 = 0$