Find the coordinates of the vertex for the parabola defined by the given quadratic function. \( f(x)=2 x^{2}+4 x+1 \) The vertex is \( \square \). (Type an ordered pair.)

To find the vertex of the parabola defined by the quadratic function \( f(x) = 2x^2 + 4x + 1 \), you can use the vertex formula \( x = -\frac{b}{2a} \). Here, \( a = 2 \) and \( b = 4 \). Plugging in those values gives: \[ x = -\frac{4}{2 \times 2} = -\frac{4}{4} = -1. \] Next, substitute \( x = -1 \) back into the function to find the \( y \)-coordinate: \[ f(-1) = 2(-1)^2 + 4(-1) + 1 = 2(1) - 4 + 1 = 2 - 4 + 1 = -1. \] Thus, the vertex is \((-1, -1)\). The vertex is \((-1, -1)\).