Derive the equation of the parabola with a focus at (0, −4) and a directrix of y = 4.

f(x) = −16x2

f(x) = 16x2

f(x) = −one sixteenth x2

f(x) = one sixteenthx2

The equation of the parabola is

Step-by-step explanation:

We start by assuming a general point on the parabola .

Using the distance formula

,

we find that the distance between and the focus (0,-4) is

, and the distance between and the directrix y =4 is . On the parabola, these distances are equal:

step-by-step explanation:

first, eliminate the parenthesis:

3

x

+

2

−

4

x

second, group like terms:

3

x

−

4

x

+

2

third, combine like terms:

(

3

−

4

)

x

+

2

→

−

1

x

+

2

→

−

x

+

2