1. **State the problem:** We have a point $P$ with coordinates $(-9, -8)$ that is dilated from the origin (center of dilation) with a scale factor of $\frac{1}{3}$. We need to find the coordinates of the image of point $P$ after this dilation. 2. **Formula for dilation:** If a point $(x, y)$ is dilated from the origin with scale factor $k$, the image coordinates $(x', y')$ are given by: $$ (x', y') = (kx, ky) $$ 3. **Apply the formula:** Given $k = \frac{1}{3}$ and $P = (-9, -8)$: $$ (x', y') = \left(\frac{1}{3} \times (-9), \frac{1}{3} \times (-8)\right) $$ 4. **Calculate each coordinate:** $$ x' = \frac{1}{3} \times (-9) = \cancel{\frac{1}{3}} \times (-9) = -3 $$ $$ y' = \frac{1}{3} \times (-8) = -\frac{8}{3} $$ 5. **Final answer:** The coordinates of the image of point $P$ after dilation are: $$ \boxed{\left(-3, -\frac{8}{3}\right)} $$