1. **State the problem:** We are given the function $$f(x) = 3 + 2 \sin\left(\frac{1}{4}x\right)$$ for $$0 \leq x \leq 2\pi$$ and asked in part (d) to describe a sequence of three transformations that transform the graph of $$y = \sin x$$ for $$0 \leq x \leq \pi$$ into the graph of $$y = f(x)$$. 2. **Recall the base function and transformations:** The base function is $$y = \sin x$$. 3. **Analyze the given function:** - The argument of sine is $$\frac{1}{4}x$$ instead of $$x$$, which means a horizontal stretch by a factor of 4. - The sine function is multiplied by 2, which means a vertical stretch by a factor of 2. - The entire function is shifted vertically upwards by 3. 4. **Sequence of transformations:** **Step 1: Horizontal stretch by factor 4** - Replace $$x$$ by $$\frac{x}{4}$$ in $$\sin x$$ to get $$\sin\left(\frac{x}{4}\right)$$. - This stretches the graph horizontally by 4 times. **Step 2: Vertical stretch by factor 2** - Multiply the function by 2 to get $$2 \sin\left(\frac{x}{4}\right)$$. - This stretches the graph vertically by a factor of 2. **Step 3: Vertical shift upwards by 3** - Add 3 to the function to get $$3 + 2 \sin\left(\frac{x}{4}\right)$$. - This shifts the entire graph up by 3 units. 5. **Order of transformations:** - First, apply the horizontal stretch. - Second, apply the vertical stretch. - Third, apply the vertical shift. This sequence transforms $$y = \sin x$$ on $$0 \leq x \leq \pi$$ to $$y = 3 + 2 \sin\left(\frac{1}{4}x\right)$$ on $$0 \leq x \leq 2\pi$$. Final answer: The graph of $$y = \sin x$$ is first stretched horizontally by a factor of 4, then stretched vertically by a factor of 2, and finally shifted vertically upwards by 3.