Consider the function \( f(x)=-2 x^{2}+8 x-7 \). a. Determine, without graphing, whether the function has a minimum value or a maxi b. Find the minimum or maximum value and determine where it occurs. c. Identify the function's domain and its range. a. The function has a

**a. Determining if the function has a minimum or maximum** We are given the quadratic function \[ f(x) = -2x^2 + 8x - 7. \] The coefficient of \(x^2\) is \(-2\), which is negative. For a quadratic function, if the coefficient of \(x^2\) is negative, the parabola opens downward. This means the function has a maximum value, not a minimum. --- **b. Finding the maximum value and where it occurs** The vertex of the parabola represents the maximum point. The \(x\)-coordinate of the vertex is given by \[ x = -\frac{b}{2a}. \] For our function, \(a = -2\) and \(b = 8\), so: \[ x = -\frac{8}{2(-2)} = -\frac{8}{-4} = 2. \] To find the \(y\)-coordinate (the maximum value), substitute \(x = 2\) into the function: \[ \begin{aligned} f(2) &= -2(2^2) + 8(2) - 7 \\ &= -2(4) + 16 - 7 \\ &= -8 + 16 - 7 \\ &= 1. \end{aligned} \] Thus, the function reaches its maximum value of \(1\) at \(x = 2\). --- **c. Identifying the domain and range** - **Domain:** A quadratic function is defined for all real numbers. Therefore, the domain is \[ (-\infty, \infty). \] - **Range:** Since the function has a maximum value of \(1\) and the parabola opens downward, the outputs \(y\) can be any number less than or equal to \(1\). Hence, the range is \[ (-\infty, 1]. \] --- **Summary of Answers:** - **a.** The function has a maximum. - **b.** The maximum value is \(1\), which occurs at \(x = 2\). - **c.** Domain: \((-\infty, \infty)\), Range: \((-\infty, 1]\).