What are the vertex, focus and directrix of # y=3 -8x -4x^2 #?

1 Answer

Vertex #(h, k)=(-1, 7)#

Focus #(h, k-p)=(-1, 7-1/16)=(-1, 111/16)#

Directrix is an equation a horizontal line

#y=k+p=7+1/16=113/16#
#y=113/16#

Explanation:

From the given equation #y=3-8x-4x^2#

Do a little rearrangement

#y=-4x^2-8x+3#

factor out -4

#y=-4(x^2+2x)+3#

Complete the square by adding 1 and subtracting 1 inside the parenthesis

#y=-4(x^2+2x+1-1)+3#

#y=-4(x+1)^2+4+3#

#y=-4(x+1)^2+7#

#y-7=-4(x+1)^2#

#(x--1)^2=-1/4(y-7)# The negative sign indicates that the parabola opens downward

#-4p=-1/4#

#p=1/16#

Vertex #(h, k)=(-1, 7)#

Focus #(h, k-p)=(-1, 7-1/16)=(-1, 111/16)#

Directrix is an equation a horizontal line

#y=k+p=7+1/16=113/16#
#y=113/16#

Kindly see the graph of #y=3-8x-4x^2#

graph{(y-3+8x+4x^2)(y-113/16)=0[-20,20,-10,10]}

God bless...I hope the explanation is useful.