To identify the value of \( k \) and whether the transformation is a compression or a stretch, we can analyze how the \( x \)-coordinate of the ordered pair changes from the original to the transformed pair.
The original pair is \( (8, 21) \) and the transformed pair is \( (2, 21) \).
The \( y \)-coordinate remains the same, which indicates no vertical transformation has occurred. We only need to focus on the \( x \)-coordinates to determine the horizontal transformation.
The transformation can be expressed as:
\[ x' = k \cdot x \]
Where:
- \( x' = 2 \) (the transformed \( x \)-coordinate)
- \( x = 8 \) (the original \( x \)-coordinate)
Plugging in the values gives us:
\[ 2 = k \cdot 8 \]
Now, solving for \( k \):
\[ k = \frac{2}{8} = \frac{1}{4} \]
Now we determine if this is a compression or a stretch:
- A \( k \) value less than 1 indicates a compression.
- A \( k \) value greater than 1 indicates a stretch.
Since \( k = \frac{1}{4} \), it indicates a compression.
The correct response is:
1/4; compression