Final Answer:
The vertex form of the given quadratic function f(x) = -x² + 2x is f(x) = -(x - 1)² + 1. Therefore, the value of 'a' in simplest form is -1.
Step-by-step explanation:
The given quadratic function is in the form f(x) = -x² + 2x. To find the vertex form, we need to complete the square. The general form for completing the square is (x - h)² + k, where (h, k) is the vertex of the parabola. In this case, we factor out the coefficient of x² from the quadratic term: -x² + 2x = -1(x² - 2x).
To complete the square, we add and subtract the square of half the coefficient of the x-term inside the parentheses: -1(x² - 2x + (2/2)² - (2/2)² ). Simplifying this expression, we get -1(x - 1)² + 1. Therefore, the vertex form is f(x) = -(x - 1)² + 1, and the value of 'a' in this form is -1.
In the vertex form, the value of 'a' determines whether the parabola opens upwards or downwards. Since 'a' is the coefficient of the squared term, a negative value indicates that the parabola opens downwards. In this case, the coefficient is -1, so the parabola opens downwards. Thus, the final answer for the value of 'a' in simplest form is -1.