Exercise 21 Page 160

The graphs of the absolute value functions $f(x)=|x|$ and $p(x)=a|x|$ both have vertices located at the origin, so we know that there haven't been any translations. However, $p(x)$ is upside down and somewhat stretched. The fact that $p(x)$ is upside down means that it has been reflected in the $x\text{-}$axis. This means $a$ is a negative number. Let's assume, temporarily, that $a=\text{-}1.$ In this case, we would only change the sign of the graph's $y\text{-}$values, taking us from the graph of $f(x)=|x|$ to the graph of $y=\text{-}|x|,$ shown below.

To get from the graph of $y=\text{-}|x|$ to the graph of $p(x),$ we need to stretch $y=\text{-}|x|$ away from the $x\text{-}$axis. Let's look at a point on each graph with the same $x\text{-}$values and compare their $y\text{-}$values.

The graph of $y=\text{-}|x|$ passes through $(1,\text{-}1)$ and the graph of $p(x)$ passes through $(1,\text{-}3).$ This means that the $y\text{-}$values in $y=\text{-}|x|$ have been stretched by a factor of $3.$ Therefore, to get both a reflection in the $x\text{-}$axis and a stretch of $3,$ we must have that $a=\text{-}3.$