1. **State the problem:** We have a translation transformation $T$ that maps point $P(-2, 2)$ to $P'(0, 0)$. We need to find the image of the point $(5, -1)$ under the same translation $T$. 2. **Recall the translation formula:** A translation moves every point by the same vector. If $T$ translates a point $(x, y)$ to $(x', y')$, then $$x' = x + a$$ $$y' = y + b$$ where $(a, b)$ is the translation vector. 3. **Find the translation vector:** Since $P(-2, 2)$ maps to $P'(0, 0)$, $$a = 0 - (-2) = 2$$ $$b = 0 - 2 = -2$$ So the translation vector is $(2, -2)$. 4. **Apply the translation to $(5, -1)$:** $$x' = 5 + 2 = 7$$ $$y' = -1 + (-2) = -3$$ 5. **Final answer:** The image of the point $(5, -1)$ under the translation $T$ is $(7, -3)$.