1. **Problem:** The curve $y = \sin x$ is transformed to $y = 4 \sin \left( \frac{1}{2} x - 30^\circ \right)$. Describe fully the sequence of transformations. 2. **Formula and rules:** - Vertical stretch by factor $a$ changes $y = f(x)$ to $y = a f(x)$. - Horizontal stretch/compression by factor $k$ changes $y = f(x)$ to $y = f(kx)$. - Horizontal shift by $h$ changes $y = f(x)$ to $y = f(x - h)$. 3. **Step 1: Identify vertical stretch** The coefficient 4 outside the sine indicates a vertical stretch by a factor of 4. 4. **Step 2: Analyze inside the sine function** Inside the sine, we have $\frac{1}{2} x - 30^\circ = \frac{1}{2} (x - 60^\circ)$ because $\frac{1}{2} x - 30^\circ = \frac{1}{2} (x - 60^\circ)$. 5. **Step 3: Horizontal stretch/compression** The factor $\frac{1}{2}$ multiplying $x$ means a horizontal stretch by a factor of $\frac{1}{\frac{1}{2}} = 2$. 6. **Step 4: Horizontal shift** The term $- 60^\circ$ inside the function means a horizontal shift to the right by $60^\circ$. 7. **Sequence of transformations:** - First, horizontally stretch the graph of $y = \sin x$ by a factor of 2. - Then, shift the graph horizontally to the right by $60^\circ$. - Finally, stretch the graph vertically by a factor of 4. **Final answer:** The transformations applied in order are: 1. Horizontal stretch by factor 2. 2. Horizontal shift right by $60^\circ$. 3. Vertical stretch by factor 4.