To rotate the point \( Q(-5, -9) \) by \( -270^\circ \) about the origin, we can convert the angle to a standard positive angle for easier calculations.
A rotation of \( -270^\circ \) is equivalent to a rotation of \( 90^\circ \) in the positive direction (since \( -270 + 360 = 90 \)).
When rotating a point \( (x, y) \) by \( 90^\circ \) counterclockwise about the origin, the new coordinates can be found using the transformation: \[ (x', y') = (-y, x) \]
Applying this transformation:
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Start with \( Q(-5, -9) \):
- \( x = -5 \)
- \( y = -9 \)
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Use the rotation formula:
- \( x' = -y = -(-9) = 9 \)
- \( y' = x = -5 \)
Thus, the new coordinates after the rotation are \( Q'(9, -5) \).
The location of \( Q' \) is \( \mathbf{Q'(9, -5)} \).