Question

fill in the information about the absolute - value functions below.
$y = \frac{1}{2}|x|$, $y = 4|x|$, $y=-1|x|$, $y = -\frac{2}{3}|x|$
(a) for each function, choose whether its graph opens upward or downward
$y = \frac{1}{2}|x|$ choose one $y = 4|x|$ choose one $y=-1|x|$ choose one $y = -\frac{2}{3}|x|$ choose one
(b) choose the equation with the narrowest graph.
$o y = \frac{1}{2}|x|$ $o y = 4|x|$ $o y=-1|x|$ $o y = -\frac{2}{3}|x|$
(c) choose the equation with the widest graph.
$o y = \frac{1}{2}|x|$ $o y = 4|x|$ $o y=-1|x|$ $o y = -\frac{2}{3}|x|$

Explanation:

Step1: Determine graph - opening direction

For absolute - value functions of the form $y = a|x|$, if $a>0$, the graph opens upward; if $a < 0$, the graph opens downward.
For $y=\frac{1}{2}|x|$, since $\frac{1}{2}>0$, it opens upward.
For $y = 4|x|$, since $4>0$, it opens upward.
For $y=-1|x|$, since $-1 < 0$, it opens downward.
For $y =-\frac{2}{3}|x|$, since $-\frac{2}{3}<0$, it opens downward.

Step2: Determine narrowest and widest graph

The magnitude of the coefficient $a$ in $y = a|x|$ determines the width of the graph. The larger the magnitude of $a$, the narrower the graph.
$| \frac{1}{2}|=\frac{1}{2}$, $|4| = 4$, $|-1| = 1$, $|-\frac{2}{3}|=\frac{2}{3}$.
Since $4>1>\frac{2}{3}>\frac{1}{2}$, the narrowest graph is $y = 4|x|$ and the widest graph is $y=\frac{1}{2}|x|$.

Answer:

(a) $y=\frac{1}{2}|x|$: upward; $y = 4|x|$: upward; $y=-1|x|$: downward; $y =-\frac{2}{3}|x|$: downward
(b) $y = 4|x|$
(c) $y=\frac{1}{2}|x|$