(a) Start with the equation \(y = x^2\). First, apply the reflection in the x-axis: \(y = -x^2\). Then apply the translation \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\), which shifts the graph 3 units to the right and 1 unit down. The equation becomes \(y = -(x-3)^2 - 1\).

(b) Start with the equation \(y = x^2\). First, apply the translation \(\begin{pmatrix} 3 \\ -1 \end{pmatrix}\), which shifts the graph 3 units to the right and 1 unit down: \(y = (x-3)^2 - 1\). Then apply the reflection in the x-axis: \(y = -(x-3)^2 + 1\).

(c) To map \(y = g(x)\) onto \(y = h(x)\), observe that \(y = g(x)\) is \(-(x-3)^2 - 1\) and \(y = h(x)\) is \(-(x-3)^2 + 1\). The transformation is a vertical translation of 2 units up, represented by \(\begin{pmatrix} 0 \\ 2 \end{pmatrix}\).