To find the coordinates of the image of point $R$ after the transformations described (a $180^\circ$ counterclockwise rotation followed by a translation 3 units to the left), we proceed step by step:

1. Identify the coordinates of point $R$.

From the graph, the coordinates of $R$ are approximately $(1, -7)$.

2. Rotate $180^\circ$ counterclockwise.

A $180^\circ$ rotation around the origin transforms any point $(x, y)$ to $(-x, -y)$.
Applying this to $R(1, -7)$:

$(x, y) \to (-x, -y)$ $(1, -7) \to (-1, 7)$

Thus, after the $180^\circ$ rotation, the new coordinates of $R$ are $(-1, 7)$.

3. Translate the point 3 units to the left.

A translation to the left by 3 units decreases the $x$-coordinate by 3 while leaving the $y$-coordinate unchanged.

Starting from $(-1, 7)$:

$(x, y) \to (x - 3, y)$ $(-1, 7) \to (-1 - 3, 7)$ $(-4, 7)$

Thus, the final coordinates of $R$ after the transformations are $(-4, 7)$.

Final Answer:

The coordinates of the image of point $R$ are $(-4, 7)$.

Do you want me to show a graphical explanation of the transformations or any further clarifications?

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