To determine the value of \( k \) in the context of transforming the ordered pair \( (8, 2) \) to \( (2, 2) \), we need to understand that such transformations often involve translations, stretches, compressions, or reflections.
We can analyze the change in the x-coordinate of the pair:
The transformation appears to involve a horizontal compression followed by a horizontal translation. The transformation from \( (8, 2) \) to \( (2, 2) \) shows that the x-coordinate has changed from \( 8 \) to \( 2 \).
To express this mathematically, we need to identify the transformation that can achieve this. One common type of transformation is of the form:
\[ (x, y) \rightarrow \left(kx + d, y\right) \]
where \( k \) is a horizontal stretch/compression factor and \( d \) is a horizontal translation.
From the original to the transformed \( x \) values: \[ 8k + d = 2 \]
If we assume there is no translation (i.e., \( d = 0 \)), it simplifies to: \[ 8k = 2, \] which gives us: \[ k = \frac{2}{8} = \frac{1}{4}. \]
However, we need to confirm any translation as well. If there's any form of translation, we can set \( d \) to some value and determine \( k \) relative to \( d \).
Let's see if \( d \) can be simplified out. If we consider the simplest case without translation:
- With \( d = 0 \): \[ 8k = 2 \Rightarrow k = \frac{1}{4}. \]
If additional translation is unknown or not specified, and we stick with uniform scaling: \[ k = \frac{1}{4} \text{ is appropriate under the assumption of no translation.} \]
Thus, the value of \( k \) that satisfies the function transformation from the original pair \( (8, 2) \) to \( (2, 2) \) is:
\[ \boxed{\frac{1}{4}}. \]