Question

Given the function f(x)=log7x
, which of the following functions is the transformation of f(x)
9 units down, 5 units left, reflected across the x-axis, and stretched vertically by a factor of 8?

Answer 1

To find the transformation of the function \( f(x) = \log_7(x) \) with the given transformations, we'll apply each transformation step by step.

1. **9 units down**: This means subtracting 9 from the function. So, we have:
\[
f(x) - 9 = \log_7(x) - 9
\]

2. **5 units left**: Moving a function left by \( h \) units involves replacing \( x \) with \( x + h \). Here \( h = 5 \), so we replace \( x \) with \( x + 5 \):
\[
\log_7(x + 5) - 9
\]

3. **Reflected across the x-axis**: To reflect a function across the x-axis, we multiply the entire function by -1. Thus, we do:
\[
-(\log_7(x + 5) - 9) = -\log_7(x + 5) + 9
\]

4. **Stretched vertically by a factor of 8**: To stretch the function vertically by a factor of \( a \), we multiply the function by \( a \). Here \( a = 8 \):
\[
8(-\log_7(x + 5) + 9) = -8\log_7(x + 5) + 72
\]

Combining these transformations yields the final function:

\[
g(x) = -8\log_7(x + 5) + 72
\]

Thus, the resulting transformed function is:

\[
g(x) = -8\log_7(x + 5) + 72
\]