A quadrilateral PQRS with vertices P(-2, 1), Q(0, 3), R(2, 2), and S(1, 0) is transformed by reflection in the x-axis, followed by a translation, and then a rotation of 180 degrees about the origin. F

The coordinate rule for reflection in the x-axis is $(x, y) \to (x, -y)$.
Given vertices $P(-2, 1)$, $Q(0, 3)$, $R(2, 2)$, and $S(1, 0)$:

$$P' = (-2, -1)$$ $$Q' = (0, -3)$$ $$R' = (2, -2)$$ $$S' = (1, 0)$$

The coordinates of $P'Q'R'S'$ are $P'(-2, -1)$, $Q'(0, -3)$, $R'(2, -2)$, and $S'(1, 0)$.

The translation vector is missing from the problem statement. For demonstration purposes, I will assume a translation vector of $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$.
The coordinate rule for this translation is $(x, y) \to (x+1, y+1)$.
Applying the rule to the vertices $P'(-2, -1)$, $Q'(0, -3)$, $R'(2, -2)$, and $S'(1, 0)$:

$$P'' = (-2+1, -1+1) = (-1, 0)$$ $$Q'' = (0+1, -3+1) = (1, -2)$$ $$R'' = (2+1, -2+1) = (3, -1)$$ $$S'' = (1+1, 0+1) = (2, 1)$$

The coordinates of $P''Q''R''S''$ are $P''(-1, 0)$, $Q''(1, -2)$, $R''(3, -1)$, and $S''(2, 1)$.

The coordinate rule for a $180^\circ$ rotation about the origin is $(x, y) \to (-x, -y)$.
Applying the rule to the vertices $P''(-1, 0)$, $Q''(1, -2)$, $R''(3, -1)$, and $S''(2, 1)$:

$$P''' = (-(-1), -(0)) = (1, 0)$$ $$Q''' = (-(1), -(-2)) = (-1, 2)$$ $$R''' = (-(3), -(-1)) = (-3, 1)$$ $$S''' = (-(2), -(1)) = (-2, -1)$$

The coordinates of $P'''Q'''R'''S'''$ are $P'''(1, 0)$, $Q'''(-1, 2)$, $R'''(-3, 1)$, and $S'''(-2, -1)$.

The coordinate rule for enlargement with scale factor $k=\frac{1}{2}$ centred at the origin is $(x, y) \to (\frac{1}{2}x, \frac{1}{2}y)$.
Applying the rule to the vertices $P'''(1, 0)$, $Q'''(-1, 2)$, $R'''(-3, 1)$, and $S'''(-2, -1)$:

$$P'''' = (\frac{1}{2} \times 1, \frac{1}{2} \times 0) = (\frac{1}{2}, 0)$$ $$Q'''' = (\frac{1}{2} \times -1, \frac{1}{2} \times 2) = (-\frac{1}{2}, 1)$$ $$R'''' = (\frac{1}{2} \times -3, \frac{1}{2} \times 1) = (-\frac{3}{2}, \frac{1}{2})$$ $$S'''' = (\frac{1}{2} \times -2, \frac{1}{2} \times -1) = (-1, -\frac{1}{2})$$

The coordinates of $P''''Q''''R''''S''''$ are $P''''(\frac{1}{2}, 0)$, $Q''''(-\frac{1}{2}, 1)$, $R''''(-\frac{3}{2}, \frac{1}{2})$, and $S''''(-1, -\frac{1}{2})$.

To create the labeled graph, plot the original quadrilateral $PQRS$ and all the transformed images $P'Q'R'S'$, $P''Q''R''S''$, $P'''Q'''R'''S'''$, and $P''''Q''''R''''S''''$ using the coordinates calculated above on a coordinate plane.

Reflection, translation, and rotation are rigid transformations, meaning they preserve the area of the shape (the area scale factor is 1).
Enlargement with a scale factor $k$ changes the area by a factor of $k^2$.
In this sequence of transformations, the only transformation that changes the area is the final enlargement with a scale factor of $k = \frac{1}{2}$.
Therefore, the area of the final image $P''''Q''''R''''S''''$ is $k^2$ times the area of the original quadrilateral $PQRS$.

$$\text{Area}(P''''Q''''R''''S'''') = \left(\frac{1}{2}\right)^2 \times \text{Area}(PQRS)$$ $$\text{Area}(P''''Q''''R''''S'''') = \frac{1}{4} \times \text{Area}(PQRS)$$

The area of $P''''Q''''R''''S''''$ is one-fourth the area of $PQRS$.