Question

select the correct answer. if ( g(x) = f(4x) ), which statement is true?
a. the graph of function ( f ) is stretched horizontally by a scale factor of 4 to create the graph of function ( g ).
b. the graph of function ( f ) is compressed vertically by a scale factor of ( \frac{1}{4} ) to create the graph of function ( g ).
c. the graph of function ( f ) is compressed horizontally by a scale factor of ( \frac{1}{4} ) to create the graph of function ( g ).
d. the graph of function ( f ) is stretched vertically by a scale factor of 4 to create the graph of function ( g ).

Explanation:

Step1: Recall Horizontal Compression/Stretch Rule

For a function transformation \( g(x) = f(kx) \), if \( |k|>1 \), the graph of \( f(x) \) is horizontally compressed by a scale factor of \( \frac{1}{k} \). Here, \( k = 4 \), so the scale factor for horizontal compression is \( \frac{1}{4} \).

Step2: Analyze Each Option

• Option A: Stretching horizontally by factor 4 would be \( g(x)=f(\frac{1}{4}x) \), not \( f(4x) \). Eliminate A.
• Option B: Vertical compression/stretch involves multiplying \( f(x) \) by a constant, not the input. \( g(x)=f(4x) \) is a horizontal transformation, not vertical. Eliminate B.
• Option C: Since \( k = 4>1 \), the graph of \( f(x) \) is horizontally compressed by \( \frac{1}{4} \) to get \( g(x)=f(4x) \). This matches.
• Option D: Vertical stretching involves multiplying \( f(x) \) by a constant, not the input. \( g(x)=f(4x) \) is horizontal, not vertical. Eliminate D.

Answer:

C. The graph of function \( f \) is compressed horizontally by a scale factor of \( \frac{1}{4} \) to create the graph of function \( g \)