https://file.aiquickdraw.com/mathbot/file/1726407592755k8rti3zs.jpg

You have provided the graph of the function \( f(x) \), and you're looking to graph \( h(x) = 2f(x) - 1 \).

### Transformation Breakdown:
1. **Vertical Stretch by a factor of 2**: The factor of 2 in \( 2f(x) \) stretches the graph of \( f(x) \) vertically by multiplying all \( y \)-values by 2.
2. **Vertical Shift down by 1**: The term \( -1 \) shifts the graph down by 1 unit.

### Applying Transformations:
- Start with \( f(x) \). The given graph seems to be piecewise linear, with key points:
   - \( (-3, 0) \)
   - \( (-2, 2) \)
   - \( (0, 2) \)
   - \( (2, 2) \)
   - \( (3, 0) \)

For each key point:
- **Stretch by 2**: Multiply the \( y \)-coordinate by 2.
- **Shift down by 1**: Subtract 1 from the \( y \)-coordinate.

Let's calculate the new points for \( h(x) = 2f(x) - 1 \):

| \( x \) | \( f(x) \) | \( 2f(x) \) | \( 2f(x) - 1 \) |
|--------|------------|-------------|----------------|
| -3     | 0          | 0           | -1             |
| -2     | 2          | 4           | 3              |
| 0      | 2          | 4           | 3              |
| 2      | 2          | 4           | 3              |
| 3      | 0          | 0           | -1             |

So, the points on the graph of \( h(x) \) are:
- \( (-3, -1) \)
- \( (-2, 3) \)
- \( (0, 3) \)
- \( (2, 3) \)
- \( (3, -1) \)

These points outline the new shape of \( h(x) \).

Would you like more details or an explanation of how to plot this? Here are a few related questions:

1. How do you identify key points on a piecewise graph?
2. How do different transformations (vertical stretch, shrink, translation) affect a graph?
3. Can horizontal shifts and stretches be combined with vertical ones?
4. How would the transformation change if the function was \( h(x) = -2f(x) + 3 \)?
5. How can you graph transformations efficiently without recalculating each point?

**Tip**: When transforming graphs, always handle stretches first before applying translations.

Graph h(x) = 2f(x) - 1, where the original graph of f(x) is provided as a piecewise linear function.

Learn how to apply vertical transformations to a piecewise linear function. This problem demonstrates stretching by a factor of 2 and shifting down by 1 to graph h(x) = 2f(x) - 1. Suitable for high school students studying function transformations.

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