What are the vertex, focus and directrix of y=3 -8x -4x^2 y=3−8x−4x2?

1 Answer
Leland Adriano Alejandro
Apr 2, 2016

Vertex (h, k)=(-1, 7)(h,k)=(−1,7)

Focus (h, k-p)=(-1, 7-1/16)=(-1, 111/16)(h,k−p)=(−1,7−116)=(−1,11116)

Directrix is an equation a horizontal line

y=k+p=7+1/16=113/16y=k+p=7+116=11316
y=113/16y=11316

Explanation:

From the given equation y=3-8x-4x^2y=3−8x−4x2

Do a little rearrangement

y=-4x^2-8x+3y=−4x2−8x+3

factor out -4

y=-4(x^2+2x)+3y=−4(x2+2x)+3

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

y=-4(x^2+2x+1-1)+3y=−4(x2+2x+1−1)+3

y=-4(x+1)^2+4+3y=−4(x+1)2+4+3

y=-4(x+1)^2+7y=−4(x+1)2+7

y-7=-4(x+1)^2y−7=−4(x+1)2

(x--1)^2=-1/4(y-7)(x−−1)2=−14(y−7) The negative sign indicates that the parabola opens downward

-4p=-1/4−4p=−14

p=1/16p=116

Vertex (h, k)=(-1, 7)(h,k)=(−1,7)

Focus (h, k-p)=(-1, 7-1/16)=(-1, 111/16)(h,k−p)=(−1,7−116)=(−1,11116)

Directrix is an equation a horizontal line

y=k+p=7+1/16=113/16y=k+p=7+116=11316
y=113/16y=11316

Kindly see the graph of y=3-8x-4x^2y=3−8x−4x2

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

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

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