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:
-
Point A (1, -1) to A' (-7, -1):
- For the x-coordinate: \[ -7 = 1 + h \implies h = -7 - 1 = -8 \]
- For the y-coordinate: \[ -1 = -1 + k \implies k = -1 - (-1) = 0 \]
-
Point B (5, -2) to B' (-1, -4):
- For the x-coordinate: \[ -1 = 5 + h \implies h = -1 - 5 = -6 \]
- For the y-coordinate: \[ -4 = -2 + k \implies k = -4 - (-2) = -2 \]
-
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.