Question

Jacinda draws triangle A(-5,1)B(-3,4)C(-4,-2) and translates it along the coordinate grid to produce triangle A'(0,2)B'(2,5)C'(1,-1). Then she 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?

The value of h is ___ and the value of k is___.

Answer 1

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.

1. **Identify the coordinates of the original triangle:**
- A(-5, 1)
- B(-3, 4)
- C(-4, -2)

2. **Identify the coordinates of the translated triangle:**
- A'(0, 2)
- B'(2, 5)
- C'(1, -1)

3. **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:
\[
2 = 1 + k \implies k = 2 - 1 = 1
\]

4. **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:
\[
\text{The value of } h \text{ is } 5 \text{ and the value of } k \text{ is } 1.
\]