Question

Write the translation functions for the translation depicted below

f(x) = x+h
g(y)=y+k

Points are:

A (1,-1)
B (5,-2)
C (2,3)
A’ (-7,-1)
B’ ( -1,-4)
C’ (0,-3)

Answer 1

To determine the translation functions based on the points provided, we can analyze the translation effects on the coordinates of each point.

The functions given are:
- \( f(x) = x + h \)
- \( g(y) = y + k \)

Where \( h \) represents the horizontal shift and \( k \) represents the vertical shift.

Let's analyze the points step by step:

1. **Point A (1, -1) to A' (-7, -1):**
- For the x-coordinate:
\[
-7 = 1 + h \implies h = -7 - 1 = -8
\]
- For the y-coordinate:
\[
-1 = -1 + k \implies k = -1 - (-1) = 0
\]

2. **Point B (5, -2) to B' (-1, -4):**
- For the x-coordinate:
\[
-1 = 5 + h \implies h = -1 - 5 = -6
\]
- For the y-coordinate:
\[
-4 = -2 + k \implies k = -4 - (-2) = -2
\]

3. **Point C (2, 3) to C' (0, -3):**
- For the x-coordinate:
\[
0 = 2 + h \implies h = 0 - 2 = -2
\]
- For the y-coordinate:
\[
-3 = 3 + k \implies k = -3 - 3 = -6
\]

Upon reviewing the translations, there's an inconsistency in the values of \( h \) and \( k \).

To find the correct translation:

From point A to A':
- \( h = -8 \), \( k = 0 \) (translation from A to A’)

From point B to B':
- \( h = -6 \), \( k = -2 \) (translation from B to B’)

From point C to C':
- \( h = -2 \), \( k = -6 \) (translation from C to C’)

Each point indicates a different translation, so it is necessary to conclude the average or most common translation values if looking for a single translation function for all points. However, since the translations are inconsistent, it would be more reasonable to calculate the translations individually.

### Summary of Translations:
- For point A to A':
- Translation function is \( f(x) = x - 8 \) and \( g(y) = y + 0 = y \) (no vertical change).

- For point B to B':
- Translation function is \( f(x) = x - 6 \) and \( g(y) = y - 2 \).

- For point C to C':
- Translation function is \( f(x) = x - 2 \) and \( g(y) = y - 6 \).

If I need to generalize (which is not relevant due to inconsistencies), we can express multiple translations, but each will guide the transformation for its respective A, B, and C points specifically.