Answer:
f(x) = (1/18)x^2 - (4/9)x - 65/18
Step-by-step explanation:
Since the focus (4,3) is higher up than is the directrix y = -6, the vertex is between y = -6 and y = 3, halfway between these values, to be exact: y = -3/2. Thus, the y-coordinate of the vertex is (-6 + 3/2), or -4.5. The x-coordinate of the vertex is 4, which comes from that 4 in (4,3). Thus, the vertex is (4, -4.5) or (4, -4 1/2).
The standard equation for a vertical parabola with vertex at (h,k) is:
4p(y-k) = (x-h)^2, where p is the distance between vertex and focus or between vertex and directrix. In this case p = 4.5. Thus, the equation of this quadratic is 4(4.5)(y - [-4.5]) = (x - 4)^2, or
18(y+4.5) = (x-4)^2, or y+4.5 = (1/18)(x-4)^2.
Let's expand this and rewrite the result as a quadratic equation in standard form y = ax^2 + bx + c:
y + 4.5 = (1/18)(x^2 - 8x + 16). Multiplying all terms by 18 to remove both fractions, we get:
18y + 81 = x^2 - 8x + 16. Rearranging these terms, we get:
18y = x^2 - 8x + 16 - 81, or
18y = x^2 - 8x - 65, or
y = f(x) = (1/18)x^2 - (4/9)x - 65/18.
