Question

Using a directrix of y = -2 and a focus of (2, 6), what quadratic function is created?

a) f(x) = -1/8(x - 2)² - 2
b) f(x) = 1/16(x - 2)² + 2
c) f(x) = 1/8(x - 2)² - 2
d) f(x) = -1/16(x + 2)² - 2

Answer 1

To find the quadratic function created using a directrix of y = -2 and a focus of (2, 6), we can use the vertex form of a quadratic function and the formula for 'a'. The quadratic function is f(x) = 1/32(x - 2)² - 2, corresponding to answer choice C.

Step-by-step explanation:

To find the quadratic function created using a directrix of y = -2 and a focus of (2, 6), we can use the vertex form of a quadratic function: f(x) = a(x - h)² + k. The vertex form of a quadratic function allows us to easily identify the vertex and determine the stretch factor (a).

The vertex form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Given that the directrix is y = -2, we know that the vertex of the parabola would be at (h, k) = (2, -2). Using the focus and the vertex, we can calculate the value of 'a' using the formula: a = 1 / (4p), where 'p' is the distance between the vertex and the focus.

In this case, the distance between the vertex (2, -2) and the focus (2, 6) is 8 (p = 8). Substituting this value into the formula, we find that a = 1 / (4 * 8) = 1 / 32. Therefore, the quadratic function created is f(x) = 1/32(x - 2)² - 2, which corresponds to answer choice C.