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{{ printedBook.courseTrack.name }} {{ printedBook.name }} The graphs of $f(x)=∣x∣$ and $g(x)=a∣x∣$ both have vertices located at the origin, so we know that there haven't been any translations. However, $g(x)$ is upside down and somewhat stretched. The fact that $g(x)$ is upside down means that it has been reflected in the $x$-axis. This means $a$ is a negative number. $a<0 $ Let's assume, temporarily, that $a=-1.$ In this case, we would only change the sign of the graph's $y$-values, taking us from the graph of $f(x)=∣x∣$ to the graph of $y=-∣x∣,$ shown below.

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

The graph of $y=-∣x∣$ passes through $(1,-1)$ and the graph of $g(x)$ passes through $(1,-4).$ This means that the $y$-values in $y=-∣x∣$ have been stretched by a factor of $4.$ $y=-∣x∣y-value -1 ×∣factorstretch 4 =p(x)=a∣x∣y-value -4 $ Therefore, to get both a reflection in the $x$-axis and a stretch of $4,$ we must have that $a=-4.$