https://file.aiquickdraw.com/mathbot/file/1728194570163xoah3mfk.jpg

The function shown in the image is \( y = |x + 5| \), which represents an absolute value function. Here's how to approach graphing it:

### Steps to Graph \( y = |x + 5| \):

1. **Understand the absolute value transformation:**
   - The basic form of an absolute value function is \( y = |x| \), which creates a "V"-shaped graph with the vertex at the origin (0, 0).
   - In \( y = |x + 5| \), the graph is shifted horizontally by 5 units to the left. This shift is due to the \( +5 \) inside the absolute value function. So the vertex of the graph will be at \( (-5, 0) \).

2. **Identify key points:**
   - Vertex: \( (-5, 0) \).
   - The graph opens upwards, as the absolute value function always outputs positive values for \( y \).

3. **Shape of the graph:**
   - For \( x < -5 \), the function behaves like \( y = -(x + 5) \) because the expression inside the absolute value is negative.
   - For \( x \geq -5 \), the function behaves like \( y = x + 5 \).

4. **Draw key points:**
   - Plot the vertex at \( (-5, 0) \).
   - Plot additional points on either side of the vertex, such as \( (-6, 1) \) and \( (-4, 1) \), and connect them to form the "V" shape.

5. **Symmetry:**
   - The graph is symmetric around the vertical line \( x = -5 \).

Would you like to see a plotted graph of this function, or do you need more details on the steps?

### Related Questions:
1. How do you determine the domain and range of an absolute value function?
2. What happens to the graph of \( y = |x + 5| \) if the constant inside changes (e.g., \( |x - 5| \))?
3. How does multiplying an absolute value function by a constant (e.g., \( y = 2|x + 5| \)) affect its graph?
4. What is the difference between horizontal and vertical shifts in absolute value functions?
5. How can you graph the inverse of an absolute value function?

**Tip:** When graphing functions, it's useful to first identify any shifts (horizontal or vertical) from the base function, as this simplifies the graphing process.

Learn how to graph the absolute value function y = |x + 5|. This lesson covers the steps to graph transformations, including horizontal shifts, and explains the shape and key points of the function. Suitable for Grades 8-10.

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