Final answer:
After calculating the constant 'a' using the given points and converting the vertex form into standard form, none of the provided options matches the derived quadratic equation. There seems to be an error in the provided options or the question itself.
Step-by-step explanation:
The equation of the quadratic function with a minimum at (7,-3) and that goes through the point (9,9) can be found by recognizing that the vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, the vertex is given as (7, -3), so the quadratic function will initially look like f(x) = a(x - 7)^2 - 3. To find the value of 'a', we can use the point (9,9) that lies on the parabola by plugging it into the equation: 9 = a(9 - 7)^2 - 3. Solving for 'a', we get a = 6, which results in the quadratic equation f(x) = 6(x - 7)^2 - 3. Expanding this and putting it into standard form ax^2 + bx + c, we arrive at f(x) = 6x^2 - 84x + 291, which does not match any of the options provided. Therefore, there might be an error in the question or the provided options.