Based on the equation provided for $g(x) = \sin(x - 45^\circ)$ and the new equation $h(x) = -\sin(x) + 1$, the transformation from $g(x)$ to $h(x)$ involves the following steps:

1. Horizontal Shift:

• $g(x) = \sin(x - 45^\circ)$ represents a sine function shifted 45 degrees to the right.
• The equation $h(x) = -\sin(x) + 1$ has no horizontal shift, so the function shifts back 45 degrees to the left to align with $\sin(x)$.

2. Reflection:

• The negative sign in $h(x) = -\sin(x) + 1$ indicates a reflection across the x-axis. This inverts the sine wave.

3. Vertical Shift:

• The $+1$ 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 45 degrees to the left (to remove the $-45^\circ$ shift).
2. Reflect the function across the x-axis.
3. Shift 1 unit upwards.

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

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

1. What are the new x-intercepts and y-intercepts of $h(x)$?
2. How does the maximum and minimum value of $g(x)$ compare to $h(x)$?
3. How would the period of the function change if the frequency of $g(x)$ were different?
4. What effect does each transformation have on the phase shift of the sine wave?
5. How would the graph look if $h(x) = -\sin(x) - 1$ instead?

Tip: Always apply horizontal shifts before reflections and vertical shifts when transforming trigonometric functions.