Let's analyze the transformation of the function [tex]\( f(x) = x^2 \)[/tex] to [tex]\( h(x) = \frac{1}{6} x^2 - 9 \)[/tex].
### Step-by-Step Analysis:
1. Identify the Original Function:
The original function is [tex]\( f(x) = x^2 \)[/tex].
2. Vertical Compression:
The term [tex]\(\frac{1}{6} x^2\)[/tex] indicates that the graph is vertically compressed by a factor.
Vertical compression by a factor [tex]\(a\)[/tex] means scaling the original function's y-values by [tex]\(a\)[/tex]. Here, [tex]\(a = \frac{1}{6}\)[/tex].
This means that every y-value of the original function [tex]\( f(x) = x^2 \)[/tex] is multiplied by [tex]\(\frac{1}{6}\)[/tex].
3. Vertical Shift:
The term [tex]\(- 9\)[/tex] indicates a vertical shift.
A vertical shift down by 9 units means translating every y-value of the compressed function downward by 9 units.
Thus, the transformation [tex]\( f(x) \rightarrow h(x) = \frac{1}{6} x^2 - 9 \)[/tex] can be summarized as follows:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is vertically compressed by a factor of 6 (since [tex]\(\frac{1}{6}\)[/tex] is the fraction by which it is compressed).
- The graph is then shifted down by 9 units.
### Conclusion:
After considering the vertical compression and vertical shift, the correct description of the transformation that has occurred is:
C. The graph of [tex]\( f(x) \)[/tex] is vertically compressed by a factor of 6 and shifted 9 units down.
