Use matrix multiplication to find the image of the triangle with data matrix $D=\left[\begin{array}{rrr}-4 & -2 & -5 \\ 0 & 4 & 5\end{array}\right]$ under the transformation that reflects points through the y-axis. Sketch both the original triangle and its image. The matrix after the transformation is (Simplify your answer.)

Step 1 ：The transformation that reflects points through the y-axis is represented by the matrix \(T=\left[\begin{array}{rr}-1 & 0 \\ 0 & 1\end{array}\right]\).

Step 2 ：The data matrix of the triangle is \(D=\left[\begin{array}{rrr}-4 & -2 & -5 \\ 0 & 4 & 5\end{array}\right]\).

Step 3 ：To find the image of the triangle under this transformation, we need to multiply the data matrix \(D\) by the transformation matrix \(T\).

Step 4 ：The result of the matrix multiplication is \(DT = \left[\begin{array}{rrr}4 & 2 & 5 \\ 0 & 4 & 5\end{array}\right]\).

Step 5 ：This gives us the coordinates of the vertices of the triangle after the transformation. The transformed triangle has vertices at (4,0), (2,4), and (5,5).

Step 6 ：Final Answer: The matrix after the transformation is \(\boxed{\left[\begin{array}{rrr}4 & 2 & 5 \\ 0 & 4 & 5\end{array}\right]}\).