Final answer:
The quadratic function in vertex form given the vertex (3, -3), direction of opening downward, and the point (2, -7) on the parabola is y = -4(x - 3)2 - 3.
Step-by-step explanation:
To write a quadratic function in vertex form given the vertex, direction of opening, and another point on the parabola, we can use the vertex form of a parabola's equation:
y = a(x - h)2 + k
where (h, k) is the vertex of the parabola. Since the parabola opens downward, the value of a will be negative. In this case, the vertex is given as (3, -3), so we substitute h = 3 and k = -3 into the equation to get:
y = a(x - 3)2 - 3
The point (2, -7) is on the parabola, so we can substitute x = 2 and y = -7 to solve for a:
-7 = a(2 - 3)2 - 3
-7 = a(-1)2 - 3
-7 = a - 3
So a = -4.
Now, we can write the complete quadratic function in vertex form:
y = -4(x - 3)2 - 3