Answer:
y = 2( x+5)^2 -4
Vertex (-5,-4)
directrix y = -33/8
focus (-5, -31/8)
Step-by-step explanation:
First identify the vertex, which is the minimum
Vertex = (-5,-4)
The vertex form is y = a( x-h)^2 + k
y = a( x- -5)^2 -4
y = a( x+5)^2 -4
We need to determine a
Substitute a point on the graph
(-3,4) is on the graph
4 = a( -3+5)^2 -4
4 = a( 2)^2 -4
4 = 4a -4
Add 4 to each side
8 = 4a
Divide by 4
8/4 = a
a=2
y = 2( x+5)^2 -4
To find the focus and directrix, write in standard form
4p(y-k)=(x-h)^2
y+4 = 2(x+5)^2
1/2 (y+4) = (x+5)^2
4p = 1/2
p = 1/8
The focal length is 1/8
Subtract this from the y coordinate to get the directrix
y = -4 -1/8
y = -32/8 - 1/8 = -33/8
Add this to the y coordinate to get the coordinate for the focus the x coordinate the same
(-5, -4+1/8)
(-5, -32/4+1/8)
(-5, -31/8)
