To analyze the transformation of the graph of \( f(x) = \ln x \) when it is replaced by \( -\frac{1}{7} f(x) \), we start by examining the effect of each component in the new function.
-
Negative Sign (-): The negative sign indicates a reflection across the x-axis. So, \( -f(x) \) means that the output values of the function are inverted.
-
Multiplication by \( \frac{1}{7} \): This indicates a vertical compression by a factor of 1/7. When a function \( f(x) \) is multiplied by a positive constant \( k \) (less than 1), it compresses the graph vertically. In this case, since \( f(x) \) is multiplied by \(-\frac{1}{7}\), it reflects across the x-axis and compresses the output values.
Putting these two transformations together, we can conclude that replacing \( f(x) \) with \( -\frac{1}{7} f(x) \) results in:
- A reflection across the x-axis (due to the negative sign).
- A vertical compression by a factor of \( \frac{1}{7} \) (due to the multiplication by \( \frac{1}{7} \)).
Therefore, the correct response is:
The graph reflects across the x-axis and vertically compresses by a factor of \( \frac{1}{7} \).