Question

Using a directrix of y = 5 with focus at (4, 1), what quadratic function is created?

f(x) = 1/4(x − 4)2 − 3
f(x) = 1/8(x + 4)2 − 3
f(x) = −1/8(x − 4)2 + 3
f(x) = -1/4(x + 4)2 − 3

Answer 1

C

Explanation:

From any point (x, y) on the parabola the focus and directrix are equidistant.

Using the distance formula

`√((x-4)^2+(y-1)^2)` = | y - 5 |

Squaring both sides

(x - 4)² + (y - 1)² = (y - 5)² ← distribute the factors in y

(x - 4)² + y² - 2y + 1 = y² - 10y + 25 ( subtract y² - 10y + 25 from both sides )

(x - 4)² + 8y - 24 = 0 ( subtract (x - 4)² from both sides )

8y - 24 = - (x - 4)² ← add 24 to both sides )

8y = - (x - 4)² + 24 ( divide both sides by 8 )

y = -
`(1)/(8)` (x - 4)² + 3

Hence

f(x) = -
`(1)/(8)` (x - 4)² + 3 → C