Question

Given the function f(x)=log7 x
, 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?(1 point)
Responses

g(x)=−8log7(x+5)−9
g left parenthesis x right parenthesis equals negative 8 logarithm subscript 7 baseline left parenthesis x plus 5 right parenthesis minus 9

g(x)=8log7(−x+5)−9
g left parenthesis x right parenthesis equals 8 logarithm subscript 7 baseline left parenthesis negative x plus 5 right parenthesis minus 9

g(x)=−8log7(x+9)−5
g left parenthesis x right parenthesis equals negative 8 logarithm subscript 7 baseline left parenthesis x plus 9 right parenthesis minus 5

g(x)=−8log7(x−5)−9

Answer 1

To transform the function \( f(x) = \log_7(x) \) according to the specified transformations—9 units down, 5 units left, reflected across the x-axis, and stretched vertically by a factor of 8—we will follow these steps:

1. **Reflection across the x-axis**: This changes \( f(x) \) to \( -f(x) \).
\[
f(x) \rightarrow -\log_7(x)
\]

2. **Stretching vertically by a factor of 8**: This multiplies the function by 8.
\[
-\log_7(x) \rightarrow -8\log_7(x)
\]

3. **Shifting 5 units left**: To shift left by 5, we replace \( x \) with \( x + 5 \).
\[
-8\log_7(x) \rightarrow -8\log_7(x + 5)
\]

4. **Shifting 9 units down**: This subtracts 9 from the entire function.
\[
-8\log_7(x + 5) \rightarrow -8\log_7(x + 5) - 9
\]

Putting all these transformations together, we arrive at the transformed function:
\[
g(x) = -8\log_7(x + 5) - 9
\]

Now, looking at the provided options, we find:
\[
g(x) = -8\log_7(x + 5) - 9
\]

Thus, the correct answer is:
\[
g(x) = -8\log_7(x + 5) - 9
\]