The problem asks for the image point of $(-1, -7)$ after a translation 4 units to the left and 2 units up.

Translation steps:

1. Left by 4 units: Subtract 4 from the x-coordinate. $-1 - 4 = -5$

2. Up by 2 units: Add 2 to the y-coordinate. $-7 + 2 = -5$

Image point:

The image point after the translation is $(-5, -5)$.

Would you like further details on how translations work or have any other questions? Here are some follow-up questions for deeper understanding:

1. What are translations in geometry, and how do they affect points?
2. How do translations differ from rotations and reflections?
3. Can translations change the size or shape of a figure?
4. How would the translation change if it was 4 units to the right instead?
5. Can you find the translation rule for different geometric transformations?

Tip: In translations, the shape and size of figures do not change—only the position changes.