Find an equation for the conic whose graph is shown.
A) $x ^ { 2 } = 4 ( y + 3 )$
B) $x ^ { 2 } = 3 ( y + 3 )$
C) $y ^ { 2 } = 12 ( x + 3 )$
D) $\frac { x } { 12 } = ( y + 3 ) ^ { 2 }$
E) $x ^ { 2 } = 12 ( y + 3 )$

Question

Find an equation for the conic whose graph is shown. 
A) $x ^ { 2 } = 4 ( y + 3 )$ 
B) $x ^ { 2 } = 3 ( y + 3 )$ 
C) $y ^ { 2 } = 12 ( x + 3 )$ 
D) $\frac { x } { 12 } = ( y + 3 ) ^ { 2 }$ 
E) $x ^ { 2 } = 12 ( y + 3 )$

Quizplus · Accepted Answer

The graph is a parabola that opens upwards, which means its equation will be of the form $x^2=4ay$ or $x^2=12a(y-k)$ for some constants $a$ and $k$. We can see that the vertex is at $(-3,-3)$, so the equation will be of the form $x^2=12a(y+3)$. We just need to determine the value of $a$. We can use the fact that the distance between the vertex and the focus is $a$, and in this case, the focus is $(0,-1)$. So we have:

$$a=\frac{1}{4}(4)=1$$

Substituting $a=1$ into our equation gives $x^2=12(y+3)$. Therefore, the answer is E.

Find an Equation for the Conic Whose Graph Is Shown x2=4(y+3)x ^ { 2 } = 4 ( y + 3 )x2=4(y+3)

Find an Equation for the Conic Whose Graph Is Shown $x ^ { 2 } = 4 ( y + 3 )$