Graph Parabolas with Vertices Not at the Origin
Find the vertex, focus, and directrix of the parabola with the given equation.
-$( x - 1 ) ^ { 2 } = 4 ( y + 4 )$
A) vertex: $( 1 , - 4 )$
focus: $( 1 , - 3 )$
directrix: $y = - 5$
B) vertex: $( - 1,4 )$
focus: $( - 1,5 )$
directrix: $y = 3 \quad$
C) vertex: $( - 4,1 )$
focus: $( - 4,2 )$
directrix: $y = 0$
D) vertex: $( 1 , - 4 )$
focus: $( 1 , - 5 )$
directrix: $x = - 3$

Question

Graph Parabolas with Vertices Not at the Origin
Find the vertex, focus, and directrix of the parabola with the given equation.
-$( x - 1 ) ^ { 2 } = 4 ( y + 4 )$
A) vertex: $( 1 , - 4 )$
focus: $( 1 , - 3 )$
directrix: $y = - 5$
B) vertex: $( - 1,4 )$
focus: $( - 1,5 )$
directrix: $y = 3 \quad$
C) vertex: $( - 4,1 )$
focus: $( - 4,2 )$
 directrix: $y = 0$
D) vertex: $( 1 , - 4 )$
focus: $( 1 , - 5 )$
directrix: $x = - 3$

Quizplus · Accepted Answer

We can rewrite the given equation in the standard form as $(y + 4) = \frac{1}{4}(x - 1)^2$. This is a parabola with vertex at $(1,-4)$. Since the parabola opens upwards, the focus is located one-fourth of the way from the vertex to the directrix, which is a horizontal line passing through the vertex. Thus, the focus is at $(1,-4 + \frac{1}{4}) = (1,-3)$ and the directrix is the line $y = -4 - \frac{1}{4} = -\frac{17}{4}$, which can be simplified to $y=-\frac{5}{4}$ or $y=-1.25$. Therefore, the answer is A.

Graph Parabolas with Vertices Not at the Origin Find the Vertex

Graph Parabolas with Vertices Not at the Origin
Find the Vertex