https://file.aiquickdraw.com/mathbot/file/1726806101398g1e7y827.jpg

Let's analyze the problem step by step.

The given function is \( y = -|x + 9| \).

### 1. Vertex:
The parent function for this is \( y = |x| \), and transformations have been applied to it.
- The term \( x + 9 \) indicates a horizontal shift to the left by 9 units.
- The negative sign outside the absolute value reflects the function across the x-axis.
So, the vertex of the function is at \( (-9, 0) \).

### 2. Axis of Symmetry:
Since the vertex is \( (-9, 0) \), the axis of symmetry is \( x = -9 \), because the absolute value function is symmetric around this vertical line.

### 3. Transformations:
The parent function \( y = |x| \) undergoes the following transformations:
- A horizontal translation 9 units to the left (due to \( x + 9 \)).
- A reflection across the x-axis (due to the negative sign outside the absolute value).

Let's now analyze the given options:

- **Option A**: Translates 9 units up and vertically stretched by a factor of 9. → Incorrect, as there is no vertical translation or stretching.
- **Option B**: Reflected across the x-axis and translated 9 units to the left. → This matches the description of our transformations.
- **Option C**: Translated 9 units to the left and vertically stretched by a factor of 9. → Incorrect, no vertical stretching.
- **Option D**: Reflected across the x-axis and translated 9 units up. → Incorrect, no vertical translation.
- **Option E**: Translated 9 units to the right and vertically stretched by a factor of 9. → Incorrect, wrong direction and no stretching.
- **Option F**: Reflected across the x-axis and translated 9 units down. → Incorrect, no downward translation.
- **Option G**: Reflected across the x-axis and translated 9 units to the right. → Incorrect, wrong direction.

The correct answer is **B**.

Would you like further clarification or details on any of these steps?

### Follow-up Questions:
1. How do you determine if a function is reflected across the x-axis?
2. What is the significance of the vertex in an absolute value function?
3. Can you explain how the axis of symmetry relates to the vertex?
4. How does horizontal translation affect the equation of the absolute value function?
5. What happens if the absolute value function has a coefficient other than 1 or -1?

### Tip:
When analyzing transformations, always check the impact of each term on the graph systematically—starting with reflections, then translations, and finally stretches or compressions.

Without graphing, identify the vertex, axis of symmetry, and transformations from the parent function f(x) = |x| for the function y = -|x + 9|.

Learn how to find the vertex, axis of symmetry, and transformations for the absolute value function y = -|x + 9|. Understand the horizontal shift and reflection, suitable for students in Grades 8-10.

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