To understand the effect on the graph of [tex]\( f(x) = |x| \)[/tex] when it is changed to [tex]\( g(x) = \left| \frac{1}{3}(x-1) \right| \)[/tex], we need to analyze the transformations applied to the function.
Step-by-step analysis of the transformations:
1. Horizontal Compression:
- The term [tex]\( \frac{1}{3}(x-1) \)[/tex] inside the absolute value affects the horizontal scale. Specifically, the factor [tex]\( \frac{1}{3} \)[/tex] compresses the graph horizontally by a factor of 3. This compression occurs because multiplying the variable [tex]\( x \)[/tex] by [tex]\( \frac{1}{3} \)[/tex] makes the graph wider by increasing the input range necessary to achieve the same output range.
2. Horizontal Shift:
- The term [tex]\( (x-1) \)[/tex] inside the absolute value signifies a horizontal shift. Subtracting 1 from [tex]\( x \)[/tex] shifts the graph 1 unit to the right. This is because the function reaches each value for [tex]\( g(x) \)[/tex] when the input [tex]\( x \)[/tex] is 1 unit larger than it was for [tex]\( f(x) \)[/tex].
Combining these transformations, we get that the graph of [tex]\( f(x) = |x| \)[/tex] under the transformation to [tex]\( g(x) = \left| \frac{1}{3}(x-1) \right| \)[/tex] is affected in two main ways:
- It is compressed horizontally by a factor of 3.
- It is shifted 1 unit to the right.
Therefore, the correct answer is:
D. The graph is stretched horizontally and shifted 1 unit to the right.
(Note: Compression by a factor of 3 horizontally is sometimes described as horizontal stretching by a factor of 3, meaning it takes 3 times the original horizontal distance to achieve the same height on the graph of [tex]\( g(x) \)[/tex]. This is why we refer to it as being "stretched horizontally.")
