To find the translations from point A \((-2, 3)\) to point A' \( (3, 5) \), we can analyze the changes in the x-coordinate and y-coordinate separately.
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Change in x-coordinate:
- Initial x = -2
- Final x = 3
- Change in x = 3 - (-2) = 3 + 2 = 5
- Therefore, we are translating the x-coordinate by adding 5: \( f(x) = x + 5 \)
-
Change in y-coordinate:
- Initial y = 3
- Final y = 5
- Change in y = 5 - 3 = 2
- Therefore, we are translating the y-coordinate by adding 2: \( g(y) = y + 2 \)
So the functions that give the translations are:
- \( f(x) = x + 5 \)
- \( g(y) = y + 2 \)
Now, let's look at the options:
- a. \( f(x) = x - 5 \) and \( g(y) = y - 2 \) (incorrect)
- b. \( f(x) = x + 6 \) and \( g(y) = y + 2 \) (incorrect for x)
- c. \( f(x) = x + 5 \) and \( g(y) = y + 2 \) (correct)
- d. \( f(x) = x + 2 \) and \( g(y) = y + 5 \) (incorrect)
The correct answer is c. \( f(x) = x + 5 \) and \( g(y) = y + 2 \).