Consider the following function. Complete parts (a) through (e) below. \( y=x^{2}+4 x+3 \) 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 \( \square \). (Type an integer or a fraction.) e. Use the results from parts (a)-(d) to graph the quadratic function.

\( \textbf{(a)} \) Since the coefficient of \( x^2 \) is positive, the parabola opens upward. \( \textbf{(b)} \) The vertex can be found by completing the square: \[ y = x^2 + 4x + 3 = (x^2 + 4x) + 3. \] Complete the square for \( x^2 + 4x \): \[ x^2 + 4x = \left(x^2 + 4x + 4\right) - 4 = (x+2)^2 - 4. \] Thus, \[ y = (x+2)^2 - 4 + 3 = (x+2)^2 - 1. \] The vertex is at \(\boxed{(-2,-1)}\). \( \textbf{(c)} \) To find the \( x \)-intercepts, set \( y = 0 \): \[ x^2 + 4x + 3 = 0. \] Factor the quadratic: \[ (x+1)(x+3) = 0. \] So, the solutions are \( x = -1 \) and \( x = -3 \). The \( x \)-intercepts are \(\boxed{-1, -3}\). \( \textbf{(d)} \) The \( y \)-intercept is found by letting \( x = 0 \): \[ y = 0^2 + 4(0) + 3 = 3. \] The \( y \)-intercept is \(\boxed{3}\). \( \textbf{(e)} \) To graph the quadratic function: - Plot the vertex at \((-2,-1)\). - The axis of symmetry is the vertical line \( x = -2 \). - Plot the \( x \)-intercepts at \((-3, 0)\) and \((-1, 0)\). - Plot the \( y \)-intercept at \((0, 3)\). - Reflect the \( y \)-intercept across the axis of symmetry to find another point. The point symmetric to \((0, 3)\) with respect to \( x = -2 \) is found by moving the same horizontal distance from \( x = -2 \); since \( 0 - (-2) = 2 \), reflect to get \( -2 - 2 = -4 \) and the point \((-4, 3)\). - Draw a smooth curve through these points to complete the graph of the parabola. This completes the step-by-step solution.