Question

1. rotate 180° around (-1,0) and reflect across the x - axis.

Explanation:

Step1: Define rotation rule

The rule for a 180 - degree rotation around the point $(a,b)=(-1,0)$ is $(x,y)\to(2a - x,2b - y)$. Let the coordinates of a point on the figure be $(x,y)$. After rotation, the new coordinates $(x_1,y_1)$ are given by $x_1=2\times(-1)-x=-2 - x$ and $y_1=2\times0 - y=-y$.

Step2: Define reflection rule

The rule for reflecting a point $(x_1,y_1)$ across the x - axis is $(x_1,y_1)\to(x_1,-y_1)$. After reflection across the x - axis, if the point after rotation is $(x_1,y_1)$, the final point $(x_2,y_2)$ has $x_2=x_1=-2 - x$ and $y_2=-y_1 = y$.

Step3: Apply to vertices

Assume the vertices of the original quadrilateral $RSTU$ have coordinates $R(x_R,y_R)$, $S(x_S,y_S)$, $T(x_T,y_T)$, $U(x_U,y_U)$.
For vertex $R$: After rotation, $x_{R1}=-2 - x_R$ and $y_{R1}=-y_R$. After reflection, $x_{R2}=-2 - x_R$ and $y_{R2}=y_R$.
Similarly, for vertex $S$: After rotation, $x_{S1}=-2 - x_S$ and $y_{S1}=-y_S$. After reflection, $x_{S2}=-2 - x_S$ and $y_{S2}=y_S$.
For vertex $T$: After rotation, $x_{T1}=-2 - x_T$ and $y_{T1}=-y_T$. After reflection, $x_{T2}=-2 - x_T$ and $y_{T2}=y_T$.
For vertex $U$: After rotation, $x_{U1}=-2 - x_U$ and $y_{U1}=-y_U$. After reflection, $x_{U2}=-2 - x_U$ and $y_{U2}=y_U$.
Plot the new - located vertices to get the transformed figure.

Answer:

The new figure is obtained by first rotating each vertex of the original figure 180 degrees around the point $(-1,0)$ using the rule $(x,y)\to(-2 - x,-y)$ and then reflecting the rotated vertices across the x - axis using the rule $(x,y)\to(x,-y)$. Plot the final - located vertices to draw the transformed quadrilateral.