Consider the following function. Complete parts (a) through (e) below. \( y=-2 x^{2}+8 x+4 \) The parabola opens upward. The parabola opens downward. b. Find the vertex. The vertex is \( \square \). (Type an ordered pair.) c. Use the quadratic formula to find the x-intercepts. The x-intercept(s) is/are \( \square \). (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 \( \square \). (Type an integer or a fraction.) e. Use the results from parts (a)-(d) to graph the quadratic . .

**a. Orientation of the Parabola** The function is \[ y=-2x^2+8x+4. \] Since the coefficient of \( x^2 \) is \(-2\) (which is negative), the parabola opens **downward**. --- **b. Vertex** The \( x \)-coordinate of the vertex is given by \[ x=-\frac{b}{2a}. \] Here, \( a=-2 \) and \( b=8 \), so \[ x=-\frac{8}{2(-2)}=-\frac{8}{-4}=2. \] Now, substitute \( x=2 \) into the function to find the \( y \)-coordinate: \[ y=-2(2)^2+8(2)+4=-2(4)+16+4=-8+16+4=12. \] Thus, the vertex is \[ (2,\,12). \] --- **c. \( x \)-Intercepts** To find the \( x \)-intercepts, set \( y=0 \): \[ -2x^2+8x+4=0. \] It is often easier to work with a positive leading coefficient, so multiply both sides by \(-1\): \[ 2x^2-8x-4=0. \] Divide by \(2\) to simplify: \[ x^2-4x-2=0. \] Now, apply the quadratic formula \[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}, \] with \( a=1 \), \( b=-4 \), and \( c=-2 \): Calculate the discriminant: \[ b^2-4ac=(-4)^2-4(1)(-2)=16+8=24. \] Thus, \[ x=\frac{-(-4)\pm\sqrt{24}}{2(1)}=\frac{4\pm2\sqrt{6}}{2}=2\pm\sqrt{6}. \] So the \( x \)-intercepts are \[ (2-\sqrt{6},\,0) \quad \text{and} \quad (2+\sqrt{6},\,0). \] Rounded to the nearest tenth, \( \sqrt{6}\approx2.4 \), so approximately: \[ (2-2.4,\,0)=(-0.4,\,0) \quad \text{and} \quad (2+2.4,\,0)=(4.4,\,0). \] --- **d. \( y \)-Intercept** The \( y \)-intercept is found by setting \( x=0 \): \[ y=-2(0)^2+8(0)+4=4. \] Thus, the \( y \)-intercept is \[ (0,\,4). \] --- **e. Graphing the Quadratic Function** To graph the function \( y=-2x^2+8x+4 \), plot the following key features: - **Vertex:** \( (2,\,12) \). - **Axis of Symmetry:** The line \( x=2 \). - **\( x \)-Intercepts:** \( (2-\sqrt{6},\,0) \) and \( (2+\sqrt{6},\,0) \) (approximately \((-0.4,\,0)\) and \((4.4,\,0)\)). - **\( y \)-Intercept:** \( (0,\,4) \). Since the parabola opens downward (because the coefficient of \( x^2 \) is \(-2\)), sketch a downward-opening curve using these points. The vertex is the highest point on the graph, and the curve is symmetric about the line \( x=2 \).