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This image presents a set of practice exercises on graphing functions based on transformations of the absolute value function \( f(x) = |x| \). The instructions ask to graph the given functions, compare them with \( f(x) = |x| \), and find their domain and range.

Here’s an overview of each function:

1. **\( d(x) = |x| - 4 \)**:
   - **Transformation**: This is a vertical shift downward by 4 units.
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( [-4, \infty) \)

2. **\( r(x) = |x| + 5 \)**:
   - **Transformation**: Vertical shift upward by 5 units.
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( [5, \infty) \)

3. **\( m(x) = |x| + 1 \)**:
   - **Transformation**: Vertical shift upward by 1 unit.
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( [1, \infty) \)

4. **\( v(x) = |x - 3| \)**:
   - **Transformation**: Horizontal shift to the right by 3 units.
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( [0, \infty) \)

5. **\( p(x) = \frac{1}{3} |x| \)**:
   - **Transformation**: Vertical compression by a factor of \( \frac{1}{3} \).
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( [0, \infty) \)

6. **\( j(x) = 3|x| \)**:
   - **Transformation**: Vertical stretch by a factor of 3.
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( [0, \infty) \)

7. **\( a(x) = -5|x| \)**:
   - **Transformation**: Reflection across the x-axis and vertical stretch by a factor of 5.
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( (-\infty, 0] \)

8. **\( q(x) = -\frac{3}{2}|x| \)**:
   - **Transformation**: Reflection across the x-axis and vertical compression by a factor of \( \frac{3}{2} \).
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( (-\infty, 0] \)

Let me know if you'd like any specific graphs, or further clarifications on how to plot these functions!

Here are five related questions you could try next:
1. How does the graph of \( |x| \) change when multiplied by a constant factor?
2. What happens to the graph of an absolute value function if it's shifted horizontally?
3. How would you find the vertex of any absolute value function?
4. What effect does reflecting an absolute value function across the x-axis have?
5. How do vertical shifts affect the range of a function?

**Tip**: Always check how horizontal and vertical transformations (shifts, stretches, compressions) affect both the domain and range of a function.

In Exercises 1-8, graph the function. Compare the graph to the graph of f(x) = |x|. Find the domain and range.

Learn how to graph absolute value functions and explore transformations such as vertical and horizontal shifts, stretches, and compressions. Includes step-by-step instructions to find domain and range for each function.

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