https://file.aiquickdraw.com/mathbot/file/1730463631570ejo721py.jpg

To solve this problem, let's examine the transformation that \( y = 2h(x) - 1 \) applies to the graph of \( y = h(x) \).

1. **Vertical Stretch by 2**: The term \( 2h(x) \) means that all \( y \)-values of \( h(x) \) will be doubled. This will stretch the graph vertically by a factor of 2. Points that were at \( y = 1 \), \( y = 2 \), and \( y = -1 \) will now be at \( y = 2 \), \( y = 4 \), and \( y = -2 \), respectively.

2. **Vertical Shift Down by 1**: The term \( -1 \) at the end means that all points will be shifted down by 1 unit.

### Steps for Transforming Key Points:

1. Identify key points on \( h(x) \) and apply the transformations.
   - For example, if a point on \( h(x) \) is \( (x, y) \), then on \( y = 2h(x) - 1 \), it will move to \( (x, 2y - 1) \).

2. Apply these transformations to the graph:
   - Starting from each original point on \( h(x) \), multiply the \( y \)-coordinate by 2 and subtract 1.

### Example of Key Points Transformed:
- If there is a point on \( h(x) \) at \( (0, 0) \), after the transformations it will remain at \( (0, 0) \) because \( 2 \cdot 0 - 1 = -1 \).
- If there is a point at \( (2, 2) \), it would transform to \( (2, 2 \cdot 2 - 1) = (2, 3) \).
- Similarly, you would apply this to other points on the graph.

Would you like me to provide more specific points and a sketch of the graph after transformation?

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### Related Questions

1. How does a vertical stretch affect the slope of linear segments on a graph?
2. What happens if the transformation was \( y = \frac{1}{2} h(x) - 1 \) instead?
3. How does a vertical shift affect the x-intercepts of a graph?
4. What is the effect of multiplying \( h(x) \) by a negative number in transformations?
5. How would we transform if we wanted to reflect \( h(x) \) across the x-axis?

### Tip
When transforming graphs, always start by identifying key points, and apply each transformation in sequence for accuracy.

The graph of y = h(x) is shown. Draw the graph of y = 2h(x) - 1.

Learn how to draw the transformed graph of y = 2h(x) - 1 based on the graph of y = h(x). This problem demonstrates a vertical stretch by a factor of 2 and a vertical shift down by 1. Suitable for students in Grades 10-12.

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