To find the coordinates of the image of point $Q(4, -7)$ after a 270° clockwise rotation about the point $(5, 1)$, we can follow these steps:

Step 1: Translate the point $Q$ to the origin

To simplify the rotation process, we first translate the center of rotation $(5, 1)$ to the origin by subtracting the coordinates of the center of rotation from the point $Q(4, -7)$:

$Q' = (4 - 5, -7 - 1) = (-1, -8)$

Now, the point $Q(-1, -8)$ is relative to the center of rotation $(0, 0)$.

Step 2: Rotate the point 270° clockwise

A 270° clockwise rotation is equivalent to a 90° counterclockwise rotation. The rule for a 90° counterclockwise rotation is:

$(x, y) \rightarrow (-y, x)$

So, applying this rule to $Q'(-1, -8)$:

$Q'' = (-(-8), -1) = (8, -1)$

Step 3: Translate back to the original position

Now, translate the point $Q''(8, -1)$ back to the original center of rotation $(5, 1)$ by adding the coordinates of the center of rotation to the point:

$Q_{\text{image}} = (8 + 5, -1 + 1) = (13, 0)$

Final Answer:

The coordinates of the image of point $Q(4, -7)$ after a 270° clockwise rotation about $(5, 1)$ are $\boxed{(13, 0)}$.

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Tip:

Remember that for a clockwise rotation, you can think of a 270° clockwise rotation as a 90° counterclockwise rotation for ease of computation.