Question

What is the equation of the quadratic graph with a focus of (1, -3) and a directrix of y= -1 ? Show work

Answer 1

Answer:

$y=\frac{-1}{4}(x-1)^2-2$

Step-by-step explanation:

Given : focus of (1, -3) and a directrix of y= -1

The distance between focus and directrix is 2p

The distance between (1,-3)  and y=-1  is 2

So 2p = 2  and p =1

Vertex (h,k) lies inbetween the focus and directrix

focus is (h, k-p)

given focus is (1,-3)

so h=1  and k-p = -3 ( we got p =1)

So k-1= -3  , k= -2

Hence, vertex is (1,-2), h=1  and k = -2

$a=\frac{1}{4p} =\frac{1}{4}$

Here focus at the bottom and directrix at the top

so parabola opens down

hence a= -1/4

Plug it in the equation

$y=a(x-h)^2+k$

$y=\frac{-1}{4}(x-1)^2-2$