Answer :
To determine the effect on the graph of \( f(x) = |x| \) when the function is changed to \( g(x) = \left| \frac{1}{3} (x-1) \right| \), let's carefully analyze the transformations involved step-by-step:
1. Horizontal Shift: The term \( x-1 \) inside the absolute value affects the horizontal position of the graph. Specifically, it shifts the graph 1 unit to the right. This transformation occurs because we need to counteract the -1 by adding 1 to \( x \), making the expression zero. Therefore, the effect is a horizontal shift to the right by 1 unit.
2. Horizontal Stretch/Compression: The factor \( \frac{1}{3} \) inside the absolute value affects the horizontal scaling of the graph. To understand this effect, consider how \( \left| \frac{1}{3}(x-1) \right| \) compares to \( |x| \):
- If we replace \( x \) with \( (x - 1) \), the graph shifts right by 1 unit.
- The factor \( \frac{1}{3} \) means that each \( x \) value on the original graph corresponds to \( 3 \) times that \( x \) value on the transformed graph, thus stretching the graph horizontally by a factor of \( 3 \).
Combining these two transformations leads to the graph:
- Being shifted 1 unit to the right.
- Being stretched horizontally by a factor of 3.
Therefore, the correct answer is:
C. The graph is stretched horizontally and shifted 1 unit to the right.
1. Horizontal Shift: The term \( x-1 \) inside the absolute value affects the horizontal position of the graph. Specifically, it shifts the graph 1 unit to the right. This transformation occurs because we need to counteract the -1 by adding 1 to \( x \), making the expression zero. Therefore, the effect is a horizontal shift to the right by 1 unit.
2. Horizontal Stretch/Compression: The factor \( \frac{1}{3} \) inside the absolute value affects the horizontal scaling of the graph. To understand this effect, consider how \( \left| \frac{1}{3}(x-1) \right| \) compares to \( |x| \):
- If we replace \( x \) with \( (x - 1) \), the graph shifts right by 1 unit.
- The factor \( \frac{1}{3} \) means that each \( x \) value on the original graph corresponds to \( 3 \) times that \( x \) value on the transformed graph, thus stretching the graph horizontally by a factor of \( 3 \).
Combining these two transformations leads to the graph:
- Being shifted 1 unit to the right.
- Being stretched horizontally by a factor of 3.
Therefore, the correct answer is:
C. The graph is stretched horizontally and shifted 1 unit to the right.
