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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 . .

Ask by Page Kirk. in the United States
Mar 17,2025

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**a. Orientation of the Parabola** The parabola opens **downward**. --- **b. Vertex** The vertex is \[ (2,\,12). \] --- **c. \( x \)-Intercepts** The \( x \)-intercepts are \[ (-0.4,\,0) \quad \text{and} \quad (4.4,\,0). \] --- **d. \( y \)-Intercept** The \( y \)-intercept is \[ (0,\,4). \] --- **e. Graphing the Quadratic Function** To graph \( y=-2x^2+8x+4 \), plot the vertex at \( (2,\,12) \), the \( y \)-intercept at \( (0,\,4) \), and the \( x \)-intercepts at approximately \( (-0.4,\,0) \) and \( (4.4,\,0) \). Draw a downward-opening parabola symmetric about the line \( x=2 \).

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Bonus Knowledge

The coefficient of \( x^2 \) in the function \( y = -2x^2 + 8x + 4 \) is negative, which means the parabola opens downward. Think of it like an upside-down "U" shape. To find the vertex, we can use the formula \( x = -\frac{b}{2a} \). Plugging in your coefficients, \( a = -2 \) and \( b = 8 \), you’ll find the x-coordinate of the vertex. Then substitute that back into the equation to find the y-coordinate. A quick calculation gives you the vertex at \( (2, 12) \). For the x-intercepts, set \( y = 0 \) and solve the equation \( -2x^2 + 8x + 4 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). After solving, the x-intercepts are approximately \( 0.7, 5.7 \). To find the y-intercept, simply evaluate the function at \( x = 0 \). Plugging in \( 0 \) gives you \( y = 4 \), which is your y-intercept. Now you can plot these points (vertex, x-intercepts, and y-intercept) to graph the quadratic! Enjoy making that parabola dance on your paper!

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