For each function, choose whether its graph opens upward or downward. $$ y=\frac{1}{2}|x|: $$ upward $$ \quad y=-\frac{1}{3}|x|: $$ downward $$ v \quad y=-2|x|: $$ downward $$ v $$ b) Choose the equation with the widest graph.

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To determine whether the graphs of the given functions open upward or downward, we can analyze the coefficients of the absolute value functions.

1. **For $$ y = \frac{1}{2}|x| $$**:
   - The coefficient of the absolute value is positive ($$\frac{1}{2}$$), so the graph opens **upward**.

2. **For $$ y = -\frac{1}{3}|x| $$**:
   - The coefficient of the absolute value is negative ($$-\frac{1}{3}$$), so the graph opens **downward**.

3. **For $$ y = -2|x| $$**:
   - The coefficient of the absolute value is also negative ($$-2$$), so the graph opens **downward**.

Now, summarizing the opening directions:
- $$ y = \frac{1}{2}|x|: $$ **upward**
- $$ y = -\frac{1}{3}|x|: $$ **downward**
- $$ y = -2|x|: $$ **downward**

### b) Choose the equation with the widest graph.

The width of the graph of an absolute value function $$ y = a|x| $$ is determined by the absolute value of the coefficient $$ a $$. The larger the absolute value of $$ a $$, the narrower the graph, and the smaller the absolute value of $$ a $$, the wider the graph.

- For $$ y = \frac{1}{2}|x| $$, the absolute value of the coefficient is $$ \frac{1}{2} $$.
- For $$ y = -\frac{1}{3}|x| $$, the absolute value of the coefficient is $$ \frac{1}{3} $$.
- For $$ y = -2|x| $$, the absolute value of the coefficient is $$ 2 $$.

Comparing the absolute values:
- $$ \frac{1}{2} $$ (for $$ y = \frac{1}{2}|x| $$)
- $$ \frac{1}{3} $$ (for $$ y = -\frac{1}{3}|x| $$)
- $$ 2 $$ (for $$ y = -2|x| $$)

The smallest absolute value is $$ \frac{1}{3} $$, which corresponds to $$ y = -\frac{1}{3}|x| $$. Therefore, this function has the widest graph.

### Final Answers:
- **Opening Directions**:
  - $$ y = \frac{1}{2}|x|: $$ upward
  - $$ y = -\frac{1}{3}|x|: $$ downward
  - $$ y = -2|x|: $$ downward

- **Widest Graph**: $$ y = -\frac{1}{3}|x| $$
- $$ y = \frac{1}{2}|x|: $$ upward - $$ y = -\frac{1}{3}|x|: $$ downward - $$ y = -2|x|: $$ downward - Widest graph: $$ y = -\frac{1}{3}|x| $$

For each function, choose whether its graph opens upward or downward. \( y=\frac{1}{2}|x|: \) upward \( \quad y=-\frac{1}{3}|x|: \) downward \( v \quad y=-2|x|: \) downward \( v \) b) Choose the equation with the widest graph.

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