Question

graph the image of parallelogram pqrs after a translation 3 units left and a rotation 180° counterclockwise around the origin.

Explanation:

Identify the original coordinates

Identify the vertices of the original parallelogram \(PQRS\) from the graph:

$$ P(-2, -1), \quad Q(1, -1), \quad R(0, -5), \quad S(-3, -5) $$

Apply the translation

Using the Translation knowledge point:

$$ (x', y') = (x - 3, y) $$

$$
LATEXBLOCK0
$$

Apply the rotation

Using the Coordinate Rotation knowledge point:

$$ (x'', y'') = (-x', -y') $$

$$
LATEXBLOCK1
$$

Answer:

The coordinates of the vertices of the final image parallelogram \(P''Q''R''S''\) after the translation and rotation are:

$$ P''(5, 1), \quad Q''(2, 1), \quad R''(3, 5), \quad S''(6, 5) $$