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Dibuje la gráfica de la siguiente función, en un plano cartesiano. Obtenga el periodo y la amplitud de la función.

The uploaded image contains a mathematical function. To provide a clear and accurate solution, I'll first transcribe the function and then guide you through graphing it, as well as calculating the period and amplitude.

The function appears to be:

\[
y = \sin\left(\frac{x + \pi}{2}\right)
\]

### 1. **Identifying the Components of the Function:**
   - **Amplitude**: The amplitude is the coefficient of the sine function. In this case, the amplitude is 1, as there's no multiplier in front of the sine function.
   - **Period**: The period of a sine function \( y = \sin(bx) \) is calculated using the formula \( \frac{2\pi}{|b|} \). Here, the function can be rewritten as \( y = \sin\left(\frac{1}{2}(x + \pi)\right) \). The coefficient of \(x\) is \( \frac{1}{2} \), so the period is:
     \[
     \text{Period} = \frac{2\pi}{\frac{1}{2}} = 4\pi
     \]
   - **Phase Shift**: The phase shift is determined by the horizontal translation of the function. The function is shifted to the left by \( \pi \) units.
   
### 2. **Graphing the Function:**
   - **Amplitude**: The maximum and minimum values of the function are 1 and -1, respectively.
   - **Period**: The function completes one full cycle every \( 4\pi \) units.
   - **Phase Shift**: The graph is shifted to the left by \( \pi \) units.
   - **Key Points**: The function will have key points at \( x = -\pi, -\frac{\pi}{2}, 0, \frac{\pi}{2}, \pi, \) and so forth.

To graph the function:
   - Start the wave at \( x = -\pi \).
   - Complete one cycle by the time you reach \( x = 3\pi \).
   - Mark the peaks at \( y = 1 \) and the troughs at \( y = -1 \).

### Summary:
- **Amplitude**: 1
- **Period**: \( 4\pi \)
- **Phase Shift**: \( -\pi \) (to the left)

Would you like a more detailed explanation of any part of this process or have any other questions?

Here are some related questions that could help deepen your understanding:
1. What is the effect of changing the amplitude on the graph of a sine function?
2. How does a phase shift affect the graph of a trigonometric function?
3. How can you determine the period of a cosine function?
4. What is the difference between the graphs of \( y = \sin(x) \) and \( y = \cos(x) \)?
5. How would adding a vertical shift to this function change its graph?

**Tip**: When graphing trigonometric functions, always start by identifying the amplitude, period, and phase shift to make the graphing process easier.

Learn how to graph a sine function with detailed steps on calculating its period and amplitude. Understand the impact of phase shift on the function's graph. Suitable for high school students studying trigonometry.

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