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The problem asks to graph the function \( h(x) = -3|x| - 1 \) and describe the transformations relative to its parent function \( f(x) = |x| \).

Let's break down the transformations of the function step by step:

1. **Parent Function**: The parent function is \( f(x) = |x| \), which is a V-shaped graph opening upwards with a vertex at the origin (0,0).
   
2. **Vertical Stretch by 3**: The factor of \( -3 \) outside the absolute value stretches the graph vertically by a factor of 3. This means the slopes of the V-shape will become steeper (3 times steeper, in fact).

3. **Reflection Across the X-axis**: The negative sign in front of the 3 reflects the graph across the x-axis. So, instead of opening upwards, the graph now opens downwards.

4. **Vertical Shift Down by 1 Unit**: The \( -1 \) at the end of the function causes a vertical shift down by 1 unit. This moves the vertex from \( (0,0) \) to \( (0, -1) \).

### Summary of Transformations:
- **Vertical Stretch** by a factor of 3.
- **Reflection** across the x-axis.
- **Vertical Shift** of 1 unit down.

Would you like a further explanation or visualizations for this? 

Here are some related questions:
1. How would the graph change if the coefficient was positive instead of negative?
2. What happens to the vertex if you change the constant term at the end?
3. How does a horizontal stretch or compression affect absolute value functions?
4. How would horizontal shifts appear in a graph of an absolute value function?
5. What if the absolute value expression had a linear term inside, such as \( |x - 2| \)?

**Tip**: When analyzing transformations, focus on one change at a time (vertical, horizontal, reflection) for clarity!

Graph the function h(x) = -3|x| - 1 and describe the transformations relative to its parent function f(x) = |x|.

Understand the transformations of the absolute value function h(x) = -3|x| - 1, including a vertical stretch, reflection across the x-axis, and vertical shift. Suitable for students in Grades 9-10 studying transformations of functions.

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