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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?

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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.

Four values of an absolute value function h(x) are shown in a table. The vertex is located at (0,0). Graph the function based on the table values.

Learn how to graph an absolute value function with vertex at (0,0) based on given values in a table. This problem involves understanding absolute value functions, identifying the slope, and plotting points to form a V-shaped graph. Suitable for Grades 8-10.

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