For the quadratic function \( \mathrm{f}(\mathrm{x})=-\mathrm{x}^{2}+6 \mathrm{x} \), answer parts (a) through (c) A. The y -intercept is 0 . (Type an integer or a decimal.) Bse the graphing tool to graph the function. (b) Determine the domain and the range of the function. The domain of \( f \) is \( (-\infty, \infty) \). (Type your answer in interval notation.) The range of \( f \) is ( \( -\infty, 9] \) (Type your answer in interval notation.) (c) Determine where the function is increasing and where it is decreasing The function is increasing on the interval \( \square \). (Type your answer in interval notation.)

Let's analyze the quadratic function \( f(x) = -x^2 + 6x \). ### Part A: Finding the y-intercept To find the y-intercept, we evaluate the function at \( x = 0 \): \[ f(0) = -0^2 + 6(0) = 0 \] So, the y-intercept is \( 0 \). ### Part B: Domain and Range 1. **Domain**: The function \( f(x) = -x^2 + 6x \) is a polynomial, and polynomials are defined for all real numbers. Therefore, the domain is: \[ (-\infty, \infty) \] 2. **Range**: To find the range, we first determine the vertex of the parabola. The vertex form of a quadratic function \( ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex: \[ x = -\frac{b}{2a} \] In our case, \( a = -1 \) and \( b = 6 \): \[ x = -\frac{6}{2(-1)} = 3 \] Now, we find the y-coordinate of the vertex by substituting \( x = 3 \) back into the function: \[ f(3) = -3^2 + 6(3) = -9 + 18 = 9 \] Since the parabola opens downwards (as indicated by the negative coefficient of \( x^2 \)), the maximum value of the function is \( 9 \) at \( x = 3 \). Therefore, the range of the function is: \[ (-\infty, 9] \] ### Part C: Increasing and Decreasing Intervals To determine where the function is increasing and decreasing, we can analyze the vertex and the direction of the parabola: - The function is increasing on the interval where \( x < 3 \). - The function is decreasing on the interval where \( x > 3 \). Thus, the intervals are: - The function is increasing on the interval: \[ (-\infty, 3) \] - The function is decreasing on the interval: \[ (3, \infty) \] ### Summary of Answers - A. The y-intercept is \( 0 \). - B. The domain of \( f \) is \( (-\infty, \infty) \) and the range of \( f \) is \( (-\infty, 9] \). - C. The function is increasing on the interval \( (-\infty, 3) \).