We will find that the translation is [tex]T_{3,-4}[/tex], and the image of the point (-3,-5) under this translation is (0, -9).
A general translation can be written as:
[tex]T_{a,b}[/tex]
Such that if we apply this to a general point (x, y) we get:
[tex]T_{a,b}(x, y) = (x+ a, y + b)[/tex]
So here we have a translation that when applied to the point (3, 5) it moves it to (6, 1), then we have:
[tex]T_{a,b}(3, 5) = (3 + a, 5 + b) = (6, 1)\\a = 3\\b = -4[/tex]
Then the translation is:
[tex]T_{3,-4}[/tex]
Now we apply this to the point (-3,-5) to get:
[tex]T_{3,-4}(-3, -5) = (-3 + 3, -4 - 5) = (0, -9)[/tex]
So the image of the point (-3, -5) under the same translation is (0, -9).
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