Consider the following function. Complete parts (a) through (e) below. \( y=x^{2}+8 x+12 \) The parabola opens downward. The parabola opens upward. b. Find the vertex. The vertex is (Type an ordered pair.) c. Find the \( x \)-intercepts. The \( x \)-intercepts are (Type an integer or a fraction. 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 function.

The function \( y = x^2 + 8x + 12 \) opens upward because the coefficient of \( x^2 \) is positive. To find the vertex, we can use the formula for the vertex of a parabola, \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = 8 \). This gives us \( x = -\frac{8}{2 \cdot 1} = -4 \). To find the \( y \)-coordinate, substitute \( x = -4 \) back into the original equation: \( y = (-4)^2 + 8(-4) + 12 = 16 - 32 + 12 = -4 \). Thus, the vertex is \((-4, -4)\). Next, to find the \( x \)-intercepts, set \( y = 0 \): \[ 0 = x^2 + 8x + 12. \] Factoring gives: \[ (x + 2)(x + 6) = 0. \] Thus, the \( x \)-intercepts are \( x = -2 \) and \( x = -6 \). For the \( y \)-intercept, set \( x = 0 \): \[ y = (0)^2 + 8(0) + 12 = 12. \] Therefore, the \( y \)-intercept is \( 12 \). Now to summarize: - The vertex is \( (-4, -4) \). - The \( x \)-intercepts are \( -2, -6 \). - The \( y \)-intercept is \( 12 \). You can sketch the graph of this quadratic function by plotting the vertex \((-4, -4)\), the \( x \)-intercepts \(-2\) and \(-6\), and the \( y \)-intercept \( 12 \). Draw a smooth curve through these points, ensuring it opens upward, to complete the graph! Happy graphing!