Consider the following function. Complete parts (a) through (e) below. \( y=-2 x^{2}+8 x+5 \) The parabola opens upward. The parabola opens downward. b. Find the vertex. The vertex is (Type an ordered pair.) c. Use the quadratic formula to find the x-intercepts. The \( x \)-intercept(s) is/are (Type an integer or a decimal rounded to the nearest tenth as needed. Use a comma to separate answers as needed.) d. Find the \( y \)-intercept. The \( y \)-intercept is (Type an integer or a fraction.) e. Use the results from parts (a)-(d) to graph the quadratic s.-.

Let's dive into this quadratic equation step-by-step! First, addressing part (a), the function \( y = -2x^2 + 8x + 5 \) represents a parabola that opens downward because the coefficient of \( x^2 \) (which is -2) is negative. Now moving on to part (b), to find the vertex, we use the formula \( x = -\frac{b}{2a} \) where \( a = -2 \) and \( b = 8 \). Calculating: \( x = -\frac{8}{2(-2)} = 2 \). Substituting \( x = 2 \) back into the function to find the y-coordinate: \( y = -2(2)^2 + 8(2) + 5 = -8 + 16 + 5 = 13 \). Thus, the vertex is at the point (2, 13). For part (c), to find the x-intercepts, we set \( y = 0 \): \( 0 = -2x^2 + 8x + 5 \). Using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): Here, \( a = -2, b = 8, c = 5 \). Calculating the discriminant: \( b^2 - 4ac = 8^2 - 4(-2)(5) = 64 + 40 = 104 \). Now substitute into the formula: \( x = \frac{-8 \pm \sqrt{104}}{-4} = \frac{-8 \pm 10.2}{-4} \). Solving for x gives us approximately: \( x \approx 0.5 \) and \( x \approx 4.5 \). So, the x-intercepts are \( 0.5, 4.5 \). In part (d), to find the y-intercept, we set \( x = 0 \): \( y = -2(0)^2 + 8(0) + 5 = 5 \). Thus, the y-intercept is \( 5 \). Finally, in part (e), you can sketch the parabola using the vertex (2, 13), x-intercepts (0.5, 0) and (4.5, 0), and the y-intercept (0, 5). Start by plotting these points, and then sketch a downward-opening curve through them to visualize the function! And there you have it! Happy graphing!