https://r2.erweima.ai/mathbot/image/1726187334552dqndi4g7.jpg

Write the equation of the graph.  Then give its range as an inequality. 

To solve the problems in the image, we need to determine the equations of the piecewise linear graphs and their respective ranges.

### Step-by-step Analysis:

1. **Graph 13:**
   - The graph is a "V" shape, indicating an absolute value function.
   - The vertex of the "V" is at \((-1, 2)\).
   - The slopes are -1 (left side) and 1 (right side).
   
   **Equation:** \(y = |x + 1| + 2\)

   **Range:** \(y \geq 2\)

2. **Graph 14:**
   - The graph is an inverted "V" shape.
   - The vertex of the "V" is at \((-1, -3)\).
   - The slopes are 2 (left side) and -2 (right side).
   
   **Equation:** \(y = -2|x + 1| - 3\)

   **Range:** \(y \leq -3\)

3. **Graph 15:**
   - The graph is a "mountain" shape, an inverted absolute value function.
   - The vertex is at \((2, 3)\).
   - The slopes are -1 (left side) and -1 (right side).
   
   **Equation:** \(y = -|x - 2| + 3\)

   **Range:** \(y \leq 3\)

4. **Graph 16:**
   - The graph is a "V" shape, indicating an absolute value function.
   - The vertex of the "V" is at \((-4, -3)\).
   - The slopes are 1 (left side) and -1 (right side).
   
   **Equation:** \(y = |x + 4| - 3\)

   **Range:** \(y \geq -3\)

5. **Graph 17:**
   - The graph is a piecewise linear function with different slopes.
   - The graph changes slope at the vertex \((1, -4)\).
   - The left line has a slope of -2 and the right line has a slope of 1.
   
   **Equation:** \(y = \begin{cases} 
      -2(x + 1) - 1 & \text{if } x < 1 \\
      x - 4 & \text{if } x \geq 1 
   \end{cases}\)

   **Range:** \(y \geq -5\)

6. **Graph 18:**
   - The graph is a "W" shape, indicating two connected absolute value functions.
   - There are two vertices: \((-1, 0)\) and \((1, 0)\).
   - The slopes are 1, -1 on the left, and -1, 1 on the right.
   
   **Equation:** \(y = |x + 1| + |x - 1|\)

   **Range:** \(y \geq 0\)

If you'd like to go over each equation in more detail, please let me know!

### Follow-up Questions:
1. What are the characteristics of absolute value functions?
2. How can you determine the vertex of an absolute value function from its graph?
3. What is the difference between a regular absolute value graph and an inverted one?
4. How do we identify the range of a piecewise function?
5. What are some real-world applications of piecewise and absolute value functions?

### Tip:
To quickly identify the range of a function from its graph, look for the minimum or maximum y-values and whether the graph extends indefinitely or is bounded.

Write the equation of the graph. Then give its range as an inequality.

This lesson covers how to find the equations and ranges of piecewise linear and absolute value graphs. Using slope-intercept form and vertex form, students will learn to express functions and their range as inequalities.

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