How does the graph of the transformed function g left parenthesis x right parenthesis equals log subscript 5 baseline left parenthesis 4 x minus 16 right parenthesis compare to the graph of its parent function f left parenthesis x right parenthesis equals log subscript 5 baseline x?

Question

(1 point)
Responses

The transformed function has been compressed horizontally and translated 16 units to the right.
The transformed function has been compressed horizontally and translated 16 units to the right.

The transformed function has been stretched horizontally and translated 16 units to the right.
The transformed function has been stretched horizontally and translated 16 units to the right.

The transformed function has been compressed horizontally and translated 4 units to the right.
The transformed function has been compressed horizontally and translated 4 units to the right.

The transformed function has been stretched horizontally and translated 4 units to the right.

Answer 1

To analyze how the transformed function \( g(x) = \log_5(4x - 16) \) compares to its parent function \( f(x) = \log_5(x) \), we need to break down the transformation step by step.

1. Inside the function: \( 4x - 16 \)

   • The expression \( 4x \) indicates a horizontal compression by a factor of \( \frac{1}{4} \).
   • The term \( -16 \) indicates a horizontal translation (to the right) by \( 4 \) units, since we set \( 4x - 16 = 0 \) to find the vertical asymptote. This gives \( 4x = 16 \), or \( x = 4 \).

2. Combining the transformations:

   • The horizontal compression is due to the multiplicative factor of \( 4 \) applied to \( x \).
   • The horizontal translation is \( 4 \) units to the right, which can be derived from the transformation \( x \rightarrow x - 4 \).

Therefore, the correct response regarding the transformations of \( g(x) \) compared to \( f(x) \) is:

The transformed function has been compressed horizontally and translated 4 units to the right.