Given the function f(x)=log7 x

Question

, 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: \[ g(x) = -8\log_7(x + 5) - 9 \]