Sketch the graph of $$ y=4 \sin x, 0 \leq x \leq 2 \pi $$

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1. **Identify the key features of the sine function:**

- The basic sine function $$ y=\sin x $$ has an amplitude of 1 and a period of $$ 2\pi $$.
   - The graph of $$ y=\sin x $$ oscillates between $$-1$$ and $$1$$ with key points:
     - Starts at $$ (0,0) $$
     - Reaches the maximum $$ ( \frac{\pi}{2}, 1 ) $$
     - Returns to $$ ( \pi,0 ) $$
     - Reaches the minimum $$ ( \frac{3\pi}{2}, -1 ) $$
     - Returns to $$ ( 2\pi,0 ) $$

2. **Apply the amplitude transformation:**

- The given function is $$ y=4\sin x $$.
   - Multiplying by 4 scales the amplitude by 4, so the new amplitude becomes $$ 4 $$. 
   - The oscillation now is between $$-4$$ and $$4$$.

3. **Determine the transformed key points:**

Since the function is $$ y=4\sin x $$ the key points are scaled vertically by 4:
   
   - At $$ x=0 $$:
     $$
     y=4\sin(0)=0
     $$
     So, the point is $$ (0,0) $$.
   
   - At $$ x=\frac{\pi}{2} $$:
     $$
     y=4\sin\left(\frac{\pi}{2}\right)=4(1)=4
     $$
     So, the point is $$ \left(\frac{\pi}{2},4\right) $$.
     
   - At $$ x=\pi $$:
     $$
     y=4\sin(\pi)=4(0)=0
     $$
     So, the point is $$ (\pi,0) $$.
     
   - At $$ x=\frac{3\pi}{2} $$:
     $$
     y=4\sin\left(\frac{3\pi}{2}\right)=4(-1)=-4
     $$
     So, the point is $$ \left(\frac{3\pi}{2},-4\right) $$.
     
   - At $$ x=2\pi $$:
     $$
     y=4\sin(2\pi)=4(0)=0
     $$
     So, the point is $$ (2\pi,0) $$.

4. **Sketching the graph:**

- Plot the points:
     - $$ (0,0) $$
     - $$ \left(\frac{\pi}{2}, 4\right) $$
     - $$ (\pi, 0) $$
     - $$ \left(\frac{3\pi}{2}, -4\right) $$
     - $$ (2\pi, 0) $$
     
   - Draw a smooth sine wave connecting these points.
   - The curve starts at the origin, reaches its maximum at $$ \frac{\pi}{2} $$, goes back to zero at $$ \pi $$, reaches its minimum at $$ \frac{3\pi}{2} $$, and then returns to zero at $$ 2\pi $$.

5. **Graph characteristics recap:**

- **Amplitude:** $$ 4 $$
   - **Period:** $$ 2\pi $$
   - **Domain:** $$ 0 \leq x \leq 2\pi $$
   - **Range:** $$ -4 \leq y \leq 4 $$

Using these steps, the sketch of $$ y=4\sin x $$ over the interval $$ 0 \leq x \leq 2\pi $$ is accurately constructed with a sine wave that has been vertically stretched by a factor of 4.
To sketch the graph of $$ y = 4 \sin x $$ from $$ 0 $$ to $$ 2\pi $$, plot the sine wave with an amplitude of 4. Key points include: - $$ (0, 0) $$ - $$ \left(\frac{\pi}{2}, 4\right) $$ - $$ (\pi, 0) $$ - $$ \left(\frac{3\pi}{2}, -4\right) $$ - $$ (2\pi, 0) $$ Draw a smooth sine curve connecting these points, ensuring the wave oscillates between -4 and 4 over the given interval.

Sketch the graph of \( y=4 \sin x, 0 \leq x \leq 2 \pi \)

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