(b) Choose the equation with the widest graph. $$ y=3|x| \quad \text { y } y=\frac{1}{3}|x| \quad \text { y } y=-\frac{1}{4}|x| \quad \text { O } y=-4 $$

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To determine which equation has the widest graph among the given options, we need to analyze the absolute value functions.

The absolute value function, denoted as $$ |x| $$, represents the distance of a number from zero on the number line. The graph of $$ |x| $$ is a V-shaped graph with the vertex at the origin (0,0).

Now, let's analyze each equation:

1. $$ y = 3|x| $$:
   - This equation is a vertical stretch of the absolute value function by a factor of 3. The graph will be wider than the standard absolute value function.

2. $$ y = \frac{1}{3}|x| $$:
   - This equation is a vertical compression of the absolute value function by a factor of 3. The graph will be narrower than the standard absolute value function.

3. $$ y = -\frac{1}{4}|x| $$:
   - This equation is a vertical compression of the absolute value function by a factor of 4 and a reflection across the x-axis. The graph will be narrower than the standard absolute value function.

4. $$ y = -4 $$:
   - This equation is a horizontal line at y = -4. It does not involve the absolute value function and will not have a V-shaped graph.

Therefore, the equation with the widest graph is $$ y = 3|x| $$.
The equation with the widest graph is $$ y = 3|x| $$.

(b) Choose the equation with the widest graph. \[ y=3|x| \quad \text { y } y=\frac{1}{3}|x| \quad \text { y } y=-\frac{1}{4}|x| \quad \text { O } y=-4 \]

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