List the transformations from the parent function \( f(x)=\sin x \) to the following function: \( g(x)=-3 \sin \left(\frac{1}{2} x-\pi\right)+3 \)

To analyze the transformations from the parent function \( f(x) = \sin x \) to the function \( g(x) = -3 \sin \left(\frac{1}{2} x - \pi\right) + 3 \), we can break down the transformations step by step: 1. **Horizontal Stretch/Compression**: The term \( \frac{1}{2} x \) indicates a horizontal stretch. The factor \( \frac{1}{2} \) means that the function is stretched by a factor of 2. This affects the period of the sine function: - The period of \( f(x) = \sin x \) is \( 2\pi \). - The period of \( g(x) \) is \( \frac{2\pi}{\frac{1}{2}} = 4\pi \). 2. **Horizontal Shift**: The term \( -\pi \) inside the sine function indicates a horizontal shift to the right by \( \pi \) units. This is because the transformation \( \sin(kx - c) \) shifts the graph to the right by \( \frac{c}{k} \). 3. **Vertical Stretch/Compression**: The coefficient \( -3 \) in front of the sine function indicates a vertical stretch by a factor of 3 and a reflection across the x-axis. The negative sign means that the sine wave is flipped upside down. 4. **Vertical Shift**: The \( +3 \) at the end of the function indicates a vertical shift upward by 3 units. Putting all these transformations together, we can summarize the transformations from \( f(x) = \sin x \) to \( g(x) = -3 \sin \left(\frac{1}{2} x - \pi\right) + 3 \) as follows: - **Horizontal Stretch** by a factor of 2 (period changes from \( 2\pi \) to \( 4\pi \)). - **Horizontal Shift** to the right by \( \pi \) units. - **Vertical Stretch** by a factor of 3 and **Reflection** across the x-axis. - **Vertical Shift** upward by 3 units. These transformations can be summarized in the following order: 1. Horizontal stretch by a factor of 2. 2. Horizontal shift to the right by \( \pi \). 3. Reflection across the x-axis. 4. Vertical stretch by a factor of 3. 5. Vertical shift upward by 3 units.