Final answer:
To rewrite the quadratic function f(x) = x² − 18x + 85 in standard form, complete the square to get f(x) = (x - 9)² + 4. The vertex of the parabola is (9, 4).
Step-by-step explanation:
To rewrite the quadratic function f(x) = x² − 18x + 85 in standard form, we need to complete the square. The standard form of a quadratic function is f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola.
First, we factor out the coefficient of x² (which is 1 in this case, so this step is not necessary). Then, we find the value that completes the square for the x-terms. This value is (−b/2a), where a is the coefficient of x² and b is the coefficient of x. So, we have (−18/2(1)) = 9. We square this value to get 81, and add and subtract this inside the quadratic expression to keep it equivalent to the original function.
So, f(x) = (x² − 18x + 81) - 81 + 85, which simplifies to f(x) = (x - 9)² + 4. Therefore, the quadratic function in standard form is f(x) = (x - 9)² + 4, and the vertex is (x, y) = (9, 4).