Bot GPT 4-o mini To identify the factor \( k \) for the vertical compression, we need to look at how the original function \( y = |x + 6| \) has been transformed into the dilated function \( y = 12|x + 6| \).
In general, if a function \( y = f(x) \) is transformed into \( y = kf(x) \), then:
- If \( k > 1 \), it indicates a vertical stretch.
- If \( 0 < k < 1 \), it indicates a vertical compression.
In our case, the original function is \( y = |x + 6| \) and the dilated function is \( y = 12|x + 6| \).
Here, the function has been changed from \( |x + 6| \) to \( 12|x + 6| \), meaning we have multiplied by \( k = 12 \).
Since \( k = 12 \) is greater than 1, it indicates a vertical stretch rather than a compression.
However, if we were looking specifically for a vertical compression, this does not apply in this case because \( k \) resulted in a vertical stretch.
Thus, the correct factors are:
- For compression: \( k < 1 \) does not apply
- Resulting \( k \) for the function as it is given, k = 12.
Therefore, if specifically asked for "compression", it might be misleading unless stated otherwise, hence there is no applicable \( k \) for compression in this transformation because it is indeed a stretch.
So, in this context, transforming \( |x + 6| \) to \( 12|x + 6| \):
- The factor \( k \) that would indicate a change is simply \( k = 12 \).
If you meant to reflect on vertical compression you'd typically express in forms less than \( 1 \). Thus, based on your query:
\[ k = 12 \] (which indicates a stretch, so expect response to vary). If you are looking to get fractions:
Considering the vertical operation alone: There is none applicable for compression since \( 12\) does not imply reduced. Thus interpretation leads to potential reflection on intended expression if specified. Prepared for potential fraction isolation otherwise toward fractional participation.
1 answer