Final answer:
The equation of the quadratic function with vertex (-2, 6) that passes through (2, -7) is f(x) = -13/16(x + 2)² + 6.
Step-by-step explanation:
To determine the equation of a quadratic function with a given vertex and a point it passes through, we use the vertex form of a quadratic equation, which is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. In our case, the vertex is (-2, 6), so the equation starts as f(x) = a(x + 2)² + 6. We must find the value of 'a' that allows the parabola to pass through the point (2, -7).
Substituting the point (2, -7) into the equation gives us -7 = a(2 + 2)² + 6. Simplifying the equation, we get -7 = a(4)² + 6. Solving for 'a', we have a = (-7 - 6) / 16 = -13 / 16. Thus, the final equation is f(x) = -13/16(x + 2)² + 6.