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.