Find the vertex, focus, and directrix of the parabola. $( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)$
A) Vertex: $\left( - 5 , \frac { 5 } { 2 } \right) ;$ Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$
B) Vertex: $\left( \frac { 5 } { 2 } , - 5 \right) ;$ Focus: $\left( - 5 , \frac { 3 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$
C) Vertex: $\left( \frac { 5 } { 2 } , - 5 \right)$; Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$
D) Vertex: $\left( - 5 , \frac { 5 } { 2 } \right) ;$ Focus: $\left( - 5 , \frac { 3 } { 2 } \right) ;$ Directrix: $y = \frac { 7 } { 2 }$
E)Vertex: $( 0,0 ) ;$ Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 7 } { 2 }$

Question

Find the vertex, focus, and directrix of the parabola. $( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)$ 
A) Vertex: $\left( - 5 , \frac { 5 } { 2 } \right) ;$ Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$
B) Vertex: $\left( \frac { 5 } { 2 } , - 5 \right) ;$ Focus: $\left( - 5 , \frac { 3 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$
C) Vertex: $\left( \frac { 5 } { 2 } , - 5 \right)$; Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 3 } { 2 }$
D) Vertex: $\left( - 5 , \frac { 5 } { 2 } \right) ;$ Focus: $\left( - 5 , \frac { 3 } { 2 } \right) ;$ Directrix: $y = \frac { 7 } { 2 }$
E)Vertex: $( 0,0 ) ;$ Focus: $\left( - 5 , \frac { 7 } { 2 } \right) ;$ Directrix: $y = \frac { 7 } { 2 }$

Quizplus · Accepted Answer

We can rewrite the given equation as $(x+5)^2=4(y-\frac{5}{2})+0$, which is in the form $(x-h)^2=4p(y-k)$, where $(h,k)$ is the vertex and $p$ is the distance between the vertex and focus (or directrix). Comparing with the given equation, we have $h=-5$, $k=\frac{5}{2}$, and $4p=4$, which implies $p=1$. Therefore, the vertex is at $(-5,\frac{5}{2})$, the focus is at $(-5,\frac{5}{2}+1)=(-5,\frac{7}{2})$, and the directrix is a horizontal line $y=\frac{5}{2}-1=\frac{3}{2}$. So, the correct choice is $	extbf{(A)}$.

Find the Vertex, Focus, and Directrix of the Parabola (x+5)2=4(y−52)( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)(x+5)2=4(y−25​)

Find the Vertex, Focus, and Directrix of the Parabola $( x + 5 ) ^ { 2 } = 4 \left( y - \frac { 5 } { 2 } \right)$