Let's analyze the graph to determine the correct function.

Key Observations from the Graph:

1. V-shape: The graph suggests it is an absolute value function.
2. Vertex at (-6, -3): The V-shape’s point of reflection (the vertex) is located at x = -6 and y = -3.
3. Positive slope: As we move right from the vertex, the slope is positive (1), indicating the function rises linearly from the vertex.
4. Shifting: The graph is shifted left by 6 units and down by 3 units.

General Form of Absolute Value Functions:

• $y = |x - h| + k$, where:
  • $h$ shifts the graph horizontally.
  • $k$ shifts it vertically.
  • If $h$ is positive, the graph shifts left; if negative, it shifts right.

Matching with the Answer Choices:

• From the observations, the function appears to be shifted left by 6 units and down by 3 units.
• This corresponds to:
  $y = |x + 6| - 3$.

Correct Answer:

• A. $y = |x + 6| - 3$.

Would you like further explanation, or do you have any additional questions?

5 Related Questions:

1. What happens if we replace $|x + 6|$ with $|x - 6|$ in the function?
2. How do horizontal and vertical shifts affect absolute value graphs?
3. What is the difference between $|x + 6|$ and $|x - 6|$ graphically?
4. How can we determine the slope of an absolute value function?
5. How does changing the slope inside the absolute value (e.g., $y = 2|x + 6| - 3$) modify the graph?

Tip:
The vertex form of an absolute value function makes it easy to identify the vertex and any shifts directly from the equation!