The table in the problem provides points for an absolute value function $h(x)$, and we're given that the vertex is at $(0, 0)$.

Step-by-Step Solution

1. Identify the Form of the Absolute Value Function: Given that the vertex is at $(0, 0)$, the function likely has the form: $h(x) = a |x|$ where $a$ is a constant that affects the slope of the function.

2. Determine the Value of $a$: Using one of the points in the table, for example, $(-4, 1)$: $h(-4) = a | -4 | = 1$ Solving for $a$: $a \cdot 4 = 1 \Rightarrow a = \frac{1}{4}$ Thus, the function is: $h(x) = \frac{1}{4} |x|$

3. Plotting the Function: Using the identified function $h(x) = \frac{1}{4} |x|$, plot the points given in the table:

   • $(-6, 1.5)$
   • $(-4, 1)$
   • $(-2, 0.5)$
   • $(0, 0)$

   These points form a V-shape on the graph, which is typical for absolute value functions.

Would you like additional details on graphing this function or any further questions?

Here are some follow-up questions to explore further:

1. What is the general form of an absolute value function, and how does changing $a$ affect its graph?
2. How would the graph change if the function had a different vertex, say $(2, 3)$?
3. What are some other transformations that can be applied to the absolute value function $h(x) = |x|$?
4. How would you graph this function if $h(x) = \frac{1}{4} |x - 2| + 3$?
5. How can absolute value functions be used to model real-world situations?

Tip: Always plot a few points on each side of the vertex to confirm the shape of absolute value graphs.