1. **State the problem:** We need to describe fully a sequence of three transformations to transform the graph of $y=\sin x$ for $0 \leq x \leq \pi$ to the graph of $y=f(x)$, specifying the order of transformations. 2. **Recall the base function:** The base function is $y=\sin x$ defined on $0 \leq x \leq \pi$. 3. **Identify the transformations:** Suppose $f(x)$ is related to $\sin x$ by transformations such as horizontal scaling, vertical scaling, and vertical/horizontal shifts or reflections. 4. **Typical transformations include:** - Horizontal scaling: $y=\sin(bx)$ changes the period by factor $\frac{1}{b}$. - Vertical scaling: $y=a\sin x$ changes amplitude to $|a|$. - Vertical shift: $y=\sin x + c$ shifts graph up/down by $c$. - Horizontal shift: $y=\sin(x - d)$ shifts graph right by $d$. 5. **Example sequence:** - Step 1: Horizontal scaling by factor $k$ (replace $x$ by $kx$), changing period. - Step 2: Vertical scaling by factor $a$ (multiply entire function by $a$), changing amplitude. - Step 3: Vertical shift by $c$ (add $c$), moving graph up/down. 6. **Order matters:** - First apply horizontal scaling to adjust period. - Then apply vertical scaling to adjust amplitude. - Finally apply vertical shift to move graph vertically. 7. **Summary:** The sequence of transformations to get $y=f(x)$ from $y=\sin x$ on $0 \leq x \leq \pi$ is: 1. Horizontally scale the graph by factor $k$ (replace $x$ by $kx$). 2. Vertically scale the graph by factor $a$ (multiply by $a$). 3. Vertically shift the graph by $c$ (add $c$). This order ensures the correct shape and position of $f(x)$ relative to $\sin x$.