Question

What is the equation of the quadratic graph with a focus of (4, −3) and a directrix of y = −6? (1 point) Select one: a. f(x) = negative one over six times x squared plus four over three times x plus eleven over six b. f(x) = one over six times x squared minus four over three times x minus eleven over six c. f(x) = three fiftieth x squared plus eleven twenty fifth x plus three fifth d. f(x) = two fiftieth x squared plus eleven twenty fifth x plus two fifth

Answer 1

The equation of the quadratic graph with a focus of (4, -3) and a directrix of y = -6 is f(x) = x^2 - 8x - 12y + 20.

Step-by-step explanation:

The equation of the quadratic graph with a focus of (4, -3) and a directrix of y = -6 can be found using the formula for the equation of a parabola. The equation is of the form (x - h)^2 = 4p(y - k), where (h, k) is the coordinates of the focus and p is the distance between the focus and the vertex. In this case, (h, k) = (4, -3) and the directrix is y = -6. The distance between the focus and the vertex is the same as the distance between the directrix and the vertex, which is 3 units. Therefore, the equation is (x - 4)^2 = 4(3)(y + 3). Simplifying, we get:

(x - 4)^2 = 12(y + 3)

Expanding, we get:

x^2 - 8x + 16 = 12y + 36

Re-arranging, we get:

x^2 - 8x - 12y + 20 = 0

So, the equation of the quadratic graph is f(x) = x^2 - 8x - 12y + 20.