What are the slopes of the arms of the functions f(x) and p(x)?
a=-3, see solution.
Practice makes perfect
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-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 p(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 p(x) passes through (1, - 3). This means that the y-values in y=-|x| have been stretched by a factor of 3.
-1_(y-value y=-|x|)
*
3_(stretch factor)
=
-3_(y-value p(x)=a|x|)
Therefore, to get both a reflection in the x-axis and a stretch of 3, we must have that a=- 3.
