To determine the values of \( h \) and \( k \) for the translation functions \( f(x) = x + h \) and \( g(y) = y + k \), we need to find out how the coordinates of the triangle change from \( A, B, C \) to \( A', B', C' \).
Let's examine the translation of each point:
-
Point \( A(1, -4) \) translates to \( A'(0, 2) \):
- For the x-coordinate: \[ 0 = 1 + h \implies h = 0 - 1 = -1 \]
- For the y-coordinate: \[ 2 = -4 + k \implies k = 2 + 4 = 6 \]
-
Point \( B(3, -1) \) translates to \( B'(2, 3) \):
- For the x-coordinate: \[ 2 = 3 + h \implies h = 2 - 3 = -1 \]
- For the y-coordinate: \[ 3 = -1 + k \implies k = 3 + 1 = 4 \] (already consistent with previous)
-
Point \( C(4, -7) \) translates to \( C'(3, -1) \):
- For the x-coordinate: \[ 3 = 4 + h \implies h = 3 - 4 = -1 \]
- For the y-coordinate: \[ -1 = -7 + k \implies k = -1 + 7 = 6 \] (also consistent)
From these calculations, we find:
- The consistent value of \( h = -1 \).
- The consistent value of \( k = 6 \).
So, the translation functions are: \[ f(x) = x - 1 \quad \text{and} \quad g(y) = y + 6 \]
Thus, the values of \( h \) and \( k \) are:
- \( h = -1 \)
- \( k = 6 \)
The correct answer is: b. h = -1 and k = 6