Answer:
f(x) = [tex]\frac{1}{16}[/tex] (x - 2)² + 2
Step-by-step explanation:
From any point (x, y ) on the parabola the focus and the directrix are equidistant.
Using the distance formula
[tex]\sqrt{(x-2)^2+(y-6)^2}[/tex] = | y + 2 | ← square both sides
(x - 2)² + (y - 6)² = (y + 2)² ← subtract (y + 2)² from both sides
(x - 2)² + (y - 6)² - (y + 2)² = 0 ← subtract (x - 2)² from both sides
(y - 6)² - (y + 2)² = - (x - 2)² ← expand left side using FOIL and simplify
y² - 12y + 36 - y² - 4y - 4 = - (x - 2)²
- 16y + 32 = - (x - 2)² ← subtract 32 from both sides
- 16y = - (x - 2)² - 32 ← divide all terms by - 16
y = [tex]\frac{1}{16}[/tex] (x - 2)² + 2
