To determine the effect on the graph of [tex]\( f(x) = |x| \)[/tex] when it is changed to [tex]\( g(x) = |3x| + 1 \)[/tex], we need to carefully analyze how the transformations affect the original graph.
1. Horizontally Compressing the Graph:
The function [tex]\( g(x) = |3x| \)[/tex] represents a horizontal transformation of [tex]\( f(x) \)[/tex]. When the argument [tex]\( x \)[/tex] of [tex]\( f(x) \)[/tex] is multiplied by 3, it means that the graph is compressed horizontally by a factor of 3. This is because input values need to be scaled down by 1/3 to produce the same output as [tex]\( f(x) \)[/tex]. Mathematically, for every unit change in [tex]\( x \)[/tex] of [tex]\( f(x) \)[/tex], the same effect is seen by a 1/3 unit change in [tex]\( x \)[/tex] of [tex]\( g(x) \)[/tex].
2. Shifting Up the Graph:
The term [tex]\( +1 \)[/tex] added to [tex]\( |3x| \)[/tex] indicates a vertical shift of the graph. Specifically, adding 1 to the function will shift the entire graph of [tex]\( g(x) = |3x| \)[/tex] up by 1 unit.
Now, let's interpret the combined effect:
- The original graph [tex]\( f(x) = |x| \)[/tex] is transformed by first compressing horizontally (by a factor of 3), resulting in [tex]\( |3x| \)[/tex].
- Then, the graph is shifted vertically upwards by 1 unit, resulting in [tex]\( |3x| + 1 \)[/tex].
So, the correct answer is:
A. The graph is compressed horizontally and shifted up 1 unit.
