https://file.aiquickdraw.com/mathbot/file/17278933187744rs09dnm.jpg

Let's analyze the problem you provided.

The question asks for the transformations required to convert the graph of the function \( f(x) = |x| \) into the function \( h(x) = -|x + 2| \).

We start by recognizing the structure of \( h(x) \):
\[
h(x) = -|x + 2|
\]
### Transformations:
1. **Reflection about the x-axis**:
   - The negative sign outside the absolute value, \( -|x + 2| \), indicates a **reflection** of the graph of \( f(x) = |x| \) over the x-axis. So this transformation applies.
   
2. **Horizontal shift**:
   - The \( +2 \) inside the absolute value, \( |x + 2| \), indicates a horizontal shift to the **left by 2 units**. So a horizontal shift is applied.

There are no vertical shifts, stretches, or reflections about the y-axis involved.

### The correct transformations are:
- **A**: Reflection about the x-axis
- **E**: Horizontal shift (left by 2 units)

Thus, the correct selections would be:
- **A. Reflection about the x-axis**
- **E. Horizontal shift**

Let me know if you would like more details or explanations on any step.

Here are 5 related questions you might explore:
1. How would the graph change if the function was \( h(x) = |x + 2| \) instead of \( -|x + 2| \)?
2. What effect does adding a constant outside the absolute value, like \( h(x) = |x| + 3 \), have on the graph?
3. How does a vertical stretch or shrink affect an absolute value graph?
4. What is the effect of reflecting a graph about the y-axis?
5. How can you determine the vertex of the graph of \( h(x) = -|x + 2| \)?

**Tip:** Always identify the order of transformations when graphing functions, as some transformations can affect others (e.g., reflections and shifts).

What transformations are needed in order to obtain the graph of h(x) from the graph of f(x)? Select all that apply.

Learn how to apply transformations to graph the absolute value function h(x) = -|x + 2| from f(x) = |x|. This problem focuses on key transformations such as reflection over the x-axis and horizontal shifts.

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