To find the values of \( h \) and \( k \) in the functions \( f(x) = x + h \) and \( g(y) = y + k \), we need to determine the translation that occurs between the original triangle and the translated triangle.
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Identify the coordinates of the original triangle:
- A(-5, 1)
- B(-3, 4)
- C(-4, -2)
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Identify the coordinates of the translated triangle:
- A'(0, 2)
- B'(2, 5)
- C'(1, -1)
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Calculate the translation by comparing the coordinates:
Let's start with point A to determine \( h \) and \( k \):
- Original point A: (-5, 1)
- Translated point A': (0, 2)
For the x-coordinates: \[ 0 = -5 + h \implies h = 0 + 5 = 5 \]
For the y-coordinates: \[ 2 = 1 + k \implies k = 2 - 1 = 1 \]
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Confirm with another point (like B or C):
- Original point B: (-3, 4)
- Translated point B': (2, 5)
For the x-coordinates: \[ 2 = -3 + h \implies h = 2 + 3 = 5 \] For the y-coordinates: \[ 5 = 4 + k \implies k = 5 - 4 = 1 \]
- Original point C: (-4, -2)
- Translated point C': (1, -1)
For the x-coordinates: \[ 1 = -4 + h \implies h = 1 + 4 = 5 \] For the y-coordinates: \[ -1 = -2 + k \implies k = -1 + 2 = 1 \]
Since the calculations for \( h \) and \( k \) are consistent across all three points, we conclude that:
- The value of \( h \) is 5.
- The value of \( k \) is 1.
Thus, the final answers are: \[ \text{The value of } h \text{ is } 5 \text{ and the value of } k \text{ is } 1. \]