From the figure, we have the following coordinates of the vertices :
[tex]\begin{gathered} Q(-4,2) \\ R(-1,1) \\ S(-2,4) \end{gathered}[/tex]
a. Rotation of 180 degrees about the origin has a rule of :
[tex](x,y)\rightarrow(-x,-y)[/tex]
The sign of the coordinates will change from positive to negative or negative to positive.
The new coordinates will be :
[tex]\begin{gathered} Q(-4,2)\rightarrow Q^{\prime}(4,-2) \\ R(-1,1)\rightarrow R^{\prime}(1,-1) \\ S(-2,4)\rightarrow S^{\prime}(2,-4) \end{gathered}[/tex]
The graph will be :
b. Reflection across the x-axis has a rule of :
[tex](x,y)\rightarrow(x,-y)[/tex]
Only the y coordinate will change its sign.
So from the coordinates we got in a.
[tex]\begin{gathered} Q^{\prime}(4,-2)\rightarrow Q^{\doubleprime}(4,2) \\ R^{\prime}(1,-1)\rightarrow R^{\doubleprime}^{\prime}(1,1) \\ S^{\prime}(2,-4)\rightarrow S^{\doubleprime}(2,4) \end{gathered}[/tex]
The graph will be :
