<image>As shown in the figure, a parabola \( y = ax^2 + bx + c \) (\(a > 0\)) is translated upward along the y-axis by \( m \) units. The shaded area \( S \) is enclosed by the translated parabola, the original parabola, and two vertical lines passing through the points \((-2,0)\) and \((2,0)\). The functional relationship of \( S \) with respect to \( m \) is ____.