Question

Given the parent function f(x)=log8x , give the new function, g(x), with the following transformations: shifted 7 units down, shrunk by a factor of 1/5 , and reflected across the y-axis. Rewrite the function g(x) below.(2 points)

Answer 1

To transform the parent function \( f(x) = \log_8 x \) into the new function \( g(x) \), we will apply the specified transformations step by step.

1. **Reflection across the y-axis**: Reflecting \( f(x) \) across the y-axis means we replace \( x \) with \( -x \):
\[
f(-x) = \log_8 (-x)
\]

2. **Shrinking by a factor of \( \frac{1}{5} \)**: To shrink the function vertically by a factor of \( \frac{1}{5} \), we multiply the output of the function by \( \frac{1}{5} \):
\[
\frac{1}{5} f(-x) = \frac{1}{5} \log_8 (-x)
\]

3. **Shifting down 7 units**: To shift the function down by 7 units, we subtract 7 from the entire function:
\[
g(x) = \frac{1}{5} \log_8 (-x) - 7
\]

Putting it all together, the new function is:
\[
g(x) = \frac{1}{5} \log_8 (-x) - 7
\]

Thus, the final transformed function is:
\[
g(x) = \frac{1}{5} \log_8 (-x) - 7
\]