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Describe the transformation of g to form a new graph, h, with the equation ℎ(𝑥) = −sin𝑥 + 1.

To describe the transformation of the function \( g(x) = \sin(x - 45^\circ) \) into the new function \( h(x) = -\sin(x) + 1 \), we need to analyze how the original function is altered to reach the new function.

### 1. **Horizontal Shift:**
   - The original function \( g(x) = \sin(x - 45^\circ) \) represents a horizontal shift of the sine function by \( 45^\circ \) to the right.
   - The new function \( h(x) = -\sin(x) + 1 \) does **not** have this horizontal shift, so the graph shifts back \( 45^\circ \) to the left, aligning with the standard sine function \( \sin(x) \).

### 2. **Reflection:**
   - The coefficient of \( -1 \) in \( h(x) = -\sin(x) + 1 \) indicates a reflection of the sine function across the x-axis.

### 3. **Vertical Shift:**
   - The constant \( +1 \) in \( h(x) = -\sin(x) + 1 \) results in a vertical shift upward by 1 unit.

### Summary of Transformations:
To transform \( g(x) = \sin(x - 45^\circ) \) into \( h(x) = -\sin(x) + 1 \):
1. Shift the graph of \( g(x) \) **45 degrees to the left**.
2. Reflect the graph across the **x-axis**.
3. Shift the graph **1 unit upwards**.

These transformations result in the new function \( h(x) \).

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Would you like further details or have any questions? Here are some related questions to consider:

1. How do the amplitudes of \( g(x) \) and \( h(x) \) compare?
2. What are the x-intercepts and y-intercepts of \( h(x) \)?
3. How would the period of \( g(x) \) change if the frequency was altered?
4. What effect does the reflection across the x-axis have on the turning points of the sine function?
5. How would the graph of \( h(x) = -\sin(x) + 1 \) compare to \( h(x) = -\sin(x - 45^\circ) + 1 \)?

**Tip:** Understanding the order of transformations (shifts, reflections, stretches/compressions) is crucial for graphing and transforming functions.

Understand how function g(x) = sin(x - 45°) transforms into h(x) = -sin(x) + 1 through horizontal shifts, reflections across the x-axis, and vertical shifts. Learn about the impact on amplitude, intercepts, and period changes in trigonometric functions.

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