https://file.aiquickdraw.com/mathbot/file/1729102545200vjhvsx36.jpg

The equation provided is:

\[
y = 2|x + 3|
\]

This is an absolute value function, which creates a "V"-shaped graph. Let's go step by step to graph it:

### Step-by-Step Graphing Process:

1. **Basic Form**: The given equation is \( y = 2 |x + 3| \), which is in the form \( y = a|x - h| + k \), where:
   - \( a = 2 \) (this controls the slope or steepness of the "V" shape),
   - \( h = -3 \) (this is a horizontal shift; the graph is shifted 3 units to the left),
   - \( k = 0 \) (there is no vertical shift in this case).

2. **Vertex**: The vertex (the point where the graph "V" bends) occurs at \( x = -3 \). So, the vertex is at \( (-3, 0) \).

3. **Shape and Direction**: The absolute value creates two linear parts:
   - For \( x \geq -3 \), the equation behaves like \( y = 2(x + 3) \),
   - For \( x < -3 \), the equation behaves like \( y = 2(-(x + 3)) = -2(x + 3) \).

4. **Slope**: The slope is 2, meaning that for every 1 unit you move horizontally, the graph moves 2 units vertically.

### Key Points:

- **At \( x = -3 \)**: The vertex is at \( y = 0 \).
- **At \( x = -2 \)**: Plugging into the equation gives \( y = 2|-2 + 3| = 2 \times 1 = 2 \).
- **At \( x = 0 \)**: Plugging into the equation gives \( y = 2|0 + 3| = 2 \times 3 = 6 \).
- **At \( x = -4 \)**: Plugging into the equation gives \( y = 2|-4 + 3| = 2 \times 1 = 2 \).
- **At \( x = -6 \)**: Plugging into the equation gives \( y = 2|-6 + 3| = 2 \times 3 = 6 \).

### Plot:

1. Plot the vertex at \( (-3, 0) \).
2. On the right side (for \( x \geq -3 \)), plot points like \( (-2, 2) \), \( (0, 6) \), etc., with a slope of 2.
3. On the left side (for \( x < -3 \)), plot points like \( (-4, 2) \), \( (-6, 6) \), etc., with a slope of -2.

The graph will form a "V" shape, with the vertex at \( (-3, 0) \), and the lines extending upwards on both sides.

Would you like any further clarifications or help with this? Here are some follow-up questions:

1. How does the slope of 2 affect the steepness of the graph compared to a slope of 1?
2. What happens if we change the value of \( h \) in the equation?
3. How would the graph change if the absolute value term had a negative sign in front of it?
4. Can you explain why the graph forms a "V" shape instead of a smooth curve?
5. What would the graph look like if we added a constant \( k \) to the equation?

**Tip**: Always plot the vertex first when dealing with absolute value functions, as it helps anchor the graph.

Learn how to graph the absolute value function y = 2|x + 3|, which forms a V-shaped graph. This problem covers key algebraic concepts such as the absolute value equation form, slope calculation, and graph interpretation. Suitable for students in Grades 8-10.

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