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Given the following graph, find the solutions of the related absolute value equation.


The graph shown is a V-shaped graph, which typically represents an absolute value function. The vertex of the V-shape is at \((0, -16)\), indicating that the equation involves an absolute value term centered at \(x = 0\).

From the graph, it seems that the function has two branches: one moving upward to the left and the other to the right. The equation for this absolute value function would be of the form:

\[
y = a |x| + b
\]

### Step-by-step:
1. **Vertex**: The vertex of the graph is at \((0, -16)\). This suggests that the absolute value function has a shift downward by 16 units. So, \(b = -16\).

2. **Slope**: The slope of the lines moving away from the vertex is determined by looking at how much the graph rises for a given horizontal movement. From the graph:
   - From the point \((0, -16)\) to \((4, 0)\), there is a rise of 16 units and a run of 4 units. Hence, the slope is \( \frac{16}{4} = 4\).
   - So the equation becomes \( y = 4|x| - 16 \).

### Equation:
\[
y = 4 |x| - 16
\]

### Solutions for \(y = 0\):
To find the solutions, set \(y = 0\):

\[
0 = 4 |x| - 16
\]

Solve for \(x\):

\[
4 |x| = 16 \implies |x| = 4
\]

Thus, \(x = 4\) or \(x = -4\).

### Final Answer:
The solutions to the absolute value equation \( y = 0 \) are \( x = 4 \) and \( x = -4 \).

Would you like more details or have any other questions? 

Here are five relative questions to expand on this:
1. How do you determine the slope of the absolute value function from a graph?
2. What role does the vertex play in the equation of an absolute value function?
3. Can you generalize how to solve absolute value equations like this algebraically?
4. How would you solve the same equation if it were shifted horizontally instead of vertically?
5. What if the slope were negative—how would the graph and equation change?

**Tip**: When graphing absolute value functions, always focus on identifying the vertex first, as it helps in writing the equation.

Given the following graph, find the solutions of the related absolute value equation.

Discover how to solve an absolute value equation from a graph with a V-shape. Learn how to determine the vertex and calculate the slope to write the equation in the form y = a|x| + b. Suitable for Grades 9-11.

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