trigonometric functions: Given the function $$ f(x)=2 \sin (3 x)+1 $$, find the amplitude, period, and phase shift of the function.

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1. The general form of a sine function is 
   $$
   f(x)=A \sin(B(x-C)) + D,
   $$
   where:
   
   - $$ A $$ is the amplitude.
   - $$ B $$ affects the period.
   - $$ C $$ is the phase shift.
   - $$ D $$ is the vertical shift.

2. For the function 
   $$
   f(x)=2 \sin (3x)+1,
   $$
   we can identify:
   
   - $$ A=2 $$
   - $$ B=3 $$
   - There is no term subtracting from $$ x $$ inside the sine function, so $$ C=0 $$.
   - $$ D=1 $$

3. **Amplitude**:  
   The amplitude is given by $$ |A| $$. Therefore, the amplitude is 
   $$
   |2|=2.
   $$

4. **Period**:  
   The period of a sine function is calculated as 
   $$
   \frac{2\pi}{|B|}.
   $$
   With $$ B=3 $$, the period is 
   $$
   \frac{2\pi}{3}.
   $$

5. **Phase Shift**:  
   The phase shift is $$ C $$, given that the function is written in the form $$ \sin(B(x-C)) $$. Since the function is $$ \sin(3x) $$ with no horizontal translation, the phase shift is 
   $$
   0.
   $$

Thus, for the function 
$$
f(x)=2 \sin (3 x)+1,
$$
- The amplitude is $$ 2 $$.
- The period is $$ \frac{2\pi}{3} $$.
- The phase shift is $$ 0 $$.
The function $$ f(x)=2 \sin (3x)+1 $$ has: - **Amplitude**: 2 - **Period**: $$ \frac{2\pi}{3} $$ - **Phase Shift**: 0

trigonometric functions: Given the function \( f(x)=2 \sin (3 x)+1 \), find the amplitude, period, and phase shift of the function.

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