Exercise 4 Page 348

First let's draw the graphs of $y=|x|-4$ and $y=|x-4|$ on the same coordinate plane. To draw the first function, consider the following general equation, where $k$ is a real number. The graph of this equation is a vertical translation of $y=|x|$ by $k$ units. In this case, the value of $k$ is $\text{-}4.$ Therefore, the graph of $y=|x|-4$ is the graph of $y=|x|$ translated $4$ units down. To draw the second function, consider the following general equation where $h$ is a positive number. The graph of this equation is a horizontal translation of $y=|x|$ by $h$ units to the right. In this case, the value of $h$ is $4.$ Therefore, the graph of $y=|x-4|$ is the graph of $y=|x|$ translated $4$ units to the right.

Now we can compare them.

Similarities

• Both graphs have the same V-shape.
• The graphs overlap when $x\ge 4$ and are parallel when $y<0.$
• Both graphs are transformations of the function $y=|x|.$

Differences

• The minimum point of $y=|x|-4$ is at $(0,\text{-}4)$ whereas for $y=|x-4|$ it is at $(4,0).$
  
• The red graph is always non-negative. However, the blue graph is negative in the interval $\text{-}4<x<4.$
  
• The blue graph is a translation of $y=|x|$ down, and the red graph is a translation of $y=|x|$ to the right.