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The given function is \( g(x) = |x| + 5 \).

This is an absolute value function with a vertical shift. Here's how to analyze and graph it:

### Step-by-step breakdown:
1. **Absolute Value Function:**
   - The basic absolute value function is \( f(x) = |x| \), which has a "V" shape, with the vertex at the origin (0,0). It opens upward.
   
2. **Vertical Shift:**
   - The \( +5 \) at the end of the function means that the graph is shifted **up by 5 units**. So instead of the vertex being at (0, 0), it is now at **(0, 5)**.

### Key Points:
- **Vertex:** (0, 5)
- The graph will be symmetrical about the y-axis because of the absolute value.
- The slope of the "arms" of the "V" is 1 and -1 on either side of the vertex.
  
### Sketch of the Graph:
- The graph will look like a "V" shape starting at the point (0, 5) and going up.
- For \( x > 0 \), the slope is 1, meaning the graph increases as \( x \) increases.
- For \( x < 0 \), the slope is -1, meaning the graph decreases as \( x \) becomes more negative.

If you'd like, I can generate a graph for you.

Would you like more details on this or have any questions?

Here are 5 related questions to expand on this:
1. What happens to the graph if we change the vertical shift to -5?
2. How would adding a horizontal shift affect the graph?
3. Can you identify the domain and range of \( g(x) = |x| + 5 \)?
4. What if we multiply the absolute value by a constant (e.g., \( g(x) = 2|x| + 5 \))? How does this change the graph?
5. How do you solve equations involving absolute value functions, such as \( |x| + 5 = 8 \)?

**Tip:** Always look at how the absolute value shifts and stretches to quickly understand changes to its graph.

Graph the following function: g(x) = |x| + 5

Learn how to graph the absolute value function g(x) = |x| + 5. This function includes a vertical shift and symmetry about the y-axis. Suitable for high school algebra students in Grades 9-12.

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