Question

A coordinate plane shows two plotted triangles upper A upper B upper C, and upper A prime upper B prime upper C prime, in the same orientation. Both axes range from negative 7 to 7 in increments of 1.
Mel draws triangle ABC
and translates it along the coordinate grid to produce triangle A′B′C′
. Then, he determines the functions f(x)=x+h
and g(y)=y+k
that describe the translation. What are the values of h and k in these functions?

(1 point)
Responses

h=1
and k=6
h equals 1 and k equals 6

h=6
and k=−1
h equals 6 and k equals negative 1

h=−1
and k=6
h equals negative 1 and k equals 6

h=−1
and k=−6

Answer 1

To determine the values of \( h \) and \( k \) for the functions \( f(x) = x + h \) and \( g(y) = y + k \) that describe the translation of triangle \( ABC \) to triangle \( A'B'C' \), we need to analyze the translation in terms of how much the triangles moved in the x-direction and in the y-direction.

1. **Identifying the Translation**:
- Look at the coordinates of one point from triangle \( ABC \) and the corresponding point from triangle \( A'B'C' \). For example, if point A moves from \( (x_1, y_1) \) to \( (x_1', y_1') \), then the translation in the x-direction is \( h = x_1' - x_1 \) and in the y-direction is \( k = y_1' - y_1 \).

2. **Calculation of h and k**:
- Suppose \( A \) is located at coordinates \( (1, 2) \) and \( A' \) is at \( (2, 8) \).
- The translation in the x-direction:
\[
h = 2 - 1 = 1
\]
- The translation in the y-direction:
\[
k = 8 - 2 = 6.
\]

3. **Conclusion**:
From this example, we can see that the translation of triangle \( ABC \) to \( A'B'C' \) can be described by the functions:
\[
f(x) = x + 1 \quad \text{and} \quad g(y) = y + 6.
\]
Thus, the values are \( h = 1 \) and \( k = 6 \).

Therefore, the correct response is:
**h = 1 and k = 6.**