For the quadratic function \( f(x)=-x^{2}+6 x \), answer parts (a) through (c) What is the \( y \)-intercept? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. (Type an integer or a decimal.) Use the graphing is no y-intercept. A. (b) Determine the domain the function. The domain of \( f \) is ( \( -\infty, \infty \) ) (Type your answer in interval notation.) The range of \( f \) is (Type your answer in interval notation.)

To analyze the quadratic function \( f(x) = -x^2 + 6x \), let's go through each part step by step. ### (a) Finding the \( y \)-intercept The \( y \)-intercept of a function occurs when \( x = 0 \). We can find it by evaluating \( f(0) \): \[ f(0) = -0^2 + 6(0) = 0 \] So, the \( y \)-intercept is \( 0 \). **Answer:** The \( y \)-intercept is \( 0 \). ### (b) Determining the domain of the function The domain of a quadratic function is all real numbers unless there are restrictions (like division by zero or square roots of negative numbers). Since \( f(x) = -x^2 + 6x \) is a polynomial, it has no restrictions. **Answer:** The domain of \( f \) is \( (-\infty, \infty) \). ### (c) Finding the range of the function To find the range, we first need to determine the vertex of the parabola represented by the function. The vertex form of a quadratic function \( ax^2 + bx + c \) can be used to find the vertex using the formula \( x = -\frac{b}{2a} \). Here, \( a = -1 \) and \( b = 6 \): \[ x = -\frac{6}{2(-1)} = 3 \] Now, we can find the maximum value of the function 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 is \( 9 \) and the function decreases without bound as \( x \) moves away from \( 3 \). Thus, the range of the function is all values less than or equal to \( 9 \). **Answer:** The range of \( f \) is \( (-\infty, 9] \).