Answer :
To determine how the transformation [tex]\( g(x) = f(4x) \)[/tex] affects the graph of the function [tex]\( f(x) \)[/tex], observe the argument of the function:
If we have [tex]\( g(x) = f(4x) \)[/tex], the argument inside the function [tex]\( f \)[/tex] is scaled by a factor of 4. This scaling affects the horizontal axis (x-axis).
1. Effect on the x-axis: Scaling the argument inside the function by 4 means [tex]\( x \)[/tex]-values in [tex]\( g(x) \)[/tex] correspond to [tex]\(\frac{1}{4}\)[/tex] of the [tex]\( x \)[/tex]-values in [tex]\( f(x) \)[/tex]. This means every point on the graph of [tex]\( f(x) \)[/tex] is moved closer to the y-axis, effectively compressing it horizontally. Therefore, the graph of [tex]\( f(x) \)[/tex] is compressed horizontally by a factor of [tex]\(\frac{1}{4}\)[/tex].
2. Checking the options:
- Option A: This states that the graph is stretched horizontally by a scale factor of 4, which is incorrect since the graph is actually compressed, not stretched.
- Option B: This suggests a vertical compression by a scale factor of [tex]\(\frac{1}{4}\)[/tex], which is incorrect since the transformation affects the horizontal axis.
- Option C: This correctly states that the graph of [tex]\( f(x) \)[/tex] is compressed horizontally by a scale factor of [tex]\(\frac{1}{4}\)[/tex], which matches our understanding.
- Option D: This suggests a vertical stretching by a scale factor of 4, which is incorrect because we are dealing with horizontal compression, not vertical stretching.
Thus, the correct statement is:
Option C: The graph of function [tex]\( f \)[/tex] is compressed horizontally by a scale factor of [tex]\(\frac{1}{4}\)[/tex] to create the graph of function [tex]\( g \)[/tex].
Therefore, the correct answer is C.
If we have [tex]\( g(x) = f(4x) \)[/tex], the argument inside the function [tex]\( f \)[/tex] is scaled by a factor of 4. This scaling affects the horizontal axis (x-axis).
1. Effect on the x-axis: Scaling the argument inside the function by 4 means [tex]\( x \)[/tex]-values in [tex]\( g(x) \)[/tex] correspond to [tex]\(\frac{1}{4}\)[/tex] of the [tex]\( x \)[/tex]-values in [tex]\( f(x) \)[/tex]. This means every point on the graph of [tex]\( f(x) \)[/tex] is moved closer to the y-axis, effectively compressing it horizontally. Therefore, the graph of [tex]\( f(x) \)[/tex] is compressed horizontally by a factor of [tex]\(\frac{1}{4}\)[/tex].
2. Checking the options:
- Option A: This states that the graph is stretched horizontally by a scale factor of 4, which is incorrect since the graph is actually compressed, not stretched.
- Option B: This suggests a vertical compression by a scale factor of [tex]\(\frac{1}{4}\)[/tex], which is incorrect since the transformation affects the horizontal axis.
- Option C: This correctly states that the graph of [tex]\( f(x) \)[/tex] is compressed horizontally by a scale factor of [tex]\(\frac{1}{4}\)[/tex], which matches our understanding.
- Option D: This suggests a vertical stretching by a scale factor of 4, which is incorrect because we are dealing with horizontal compression, not vertical stretching.
Thus, the correct statement is:
Option C: The graph of function [tex]\( f \)[/tex] is compressed horizontally by a scale factor of [tex]\(\frac{1}{4}\)[/tex] to create the graph of function [tex]\( g \)[/tex].
Therefore, the correct answer is C.
