Question

(a) the graph of f(x)=|x| is shown below. transform it to get the graph of g(x)=|x - 1|. need help?

Explanation:

Step1: Recall Horizontal Shift Rule

For a function \( y = f(x - h) \), the graph of \( f(x) \) is shifted \( h \) units to the right if \( h>0 \), and \( |h| \) units to the left if \( h < 0 \). Here, \( g(x)=|x - 1|=f(x - 1) \) with \( f(x)=|x| \) and \( h = 1>0 \).

Step2: Apply the Shift to \( f(x) \)

The vertex of \( f(x)=|x| \) is at \( (0,0) \). When we apply the horizontal shift \( h = 1 \) (right shift), the vertex of \( g(x)=|x - 1| \) will be at \( (0 + 1,0)=(1,0) \). The shape of the absolute - value graph (a V - shape) remains the same, with the left side having a slope of \( - 1 \) and the right side having a slope of \( 1 \), but now centered at \( x = 1 \) instead of \( x = 0 \). So we take the graph of \( y = |x| \) and shift every point on it 1 unit to the right. For example, the point \( (-1,1) \) on \( y = |x| \) will move to \( (0,1) \) on \( y=|x - 1| \), the point \( (1,1) \) on \( y = |x| \) will move to \( (2,1) \) on \( y=|x - 1| \), etc.

Answer:

To graph \( g(x)=|x - 1| \) from \( f(x)=|x| \), shift the graph of \( f(x) \) 1 unit to the right. The vertex moves from \( (0,0) \) to \( (1,0) \), and the V - shaped graph retains its slope characteristics (slope - 1 on the left of \( x = 1 \), slope 1 on the right of \( x = 1 \)).