Find the amplitude​ (if one​ exists), period, and phase shift of the function. Graph the function. Be sure to label key points. Show at least two periods.
y equals 2 sine left parenthesis pi x plus 3 right parenthesis minus 2
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Part 1
What is the​ amplitude? Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The amplitude is 
  
enter your response here.
​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.)
B.
The function does not have an amplitude.
Part 2
What is the​ period?
  
enter your response here
​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.)
Part 3
What is the phase​ shift?
  
enter your response here
​(Simplify your answer. Type an exact​ answer, using pi as needed. Use integers or fractions for any numbers in the​ expression.)

We are working with the function:

\[
y = 2\sin\left(\pi x + 3\right) - 2
\]

### Part 1: Amplitude

The general form of a sine function is:

\[
y = A\sin(Bx + C) + D
\]

Where:
- \( A \) is the amplitude
- \( B \) determines the period
- \( C \) determines the phase shift
- \( D \) is the vertical shift

From the given function \( y = 2\sin(\pi x + 3) - 2 \), we see that \( A = 2 \). Therefore, the **amplitude** is:

\[
\text{Amplitude} = |A| = |2| = 2
\]

So, the correct answer is:
\[
\boxed{2}
\]

### Part 2: Period

The period of a sine function is calculated as:

\[
\text{Period} = \frac{2\pi}{|B|}
\]

In our case, \( B = \pi \). Thus, the period is:

\[
\text{Period} = \frac{2\pi}{\pi} = 2
\]

So, the **period** is:
\[
\boxed{2}
\]

### Part 3: Phase Shift

The phase shift is given by the formula:

\[
\text{Phase shift} = -\frac{C}{B}
\]

In this case, \( C = 3 \) and \( B = \pi \). Thus, the phase shift is:

\[
\text{Phase shift} = -\frac{3}{\pi}
\]

So, the **phase shift** is:
\[
\boxed{-\frac{3}{\pi}}
\]

### Graphing the Function

To graph \( y = 2\sin(\pi x + 3) - 2 \), we note the following key points:
- The amplitude is 2, meaning the graph oscillates between -4 and 0 (due to the vertical shift of -2).
- The period is 2, so one complete cycle occurs over an interval of 2 units along the x-axis.
- The phase shift is \( -\frac{3}{\pi} \), which shifts the graph to the left by that amount.

I can graph this for you if you'd like, showing at least two periods and labeling the key points.

---

Would you like to see the graph now or any additional details? Here are five related questions that might help you explore the topic further:

1. How do the amplitude and vertical shift affect the graph's range?
2. What happens to the period if we change \( B \) to \( 2\pi \)?
3. How can we identify phase shift from a cosine function?
4. How would the graph look if the function was \( \cos \) instead of \( \sin \)?
5. What effect does a negative amplitude have on the graph?

**Tip:** Always remember that a phase shift moves the graph horizontally, while a vertical shift moves it up or down.

Find the amplitude, period, and phase shift of the function. Graph the function. Be sure to label key points. Show at least two periods. y = 2 sin(πx + 3) - 2

Learn how to find the amplitude, period, and phase shift of the function y = 2 sin(πx + 3) - 2. This problem involves key trigonometric concepts, such as the general sine wave equation, graphing techniques, and understanding phase shifts. Suitable for high school students in Grades 10-12.

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