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Determinar: 
● El periodo 
● La amplitud 
● El rango 
● El dominio 
De las siguientes funciones

Let's analyze the graphs of the two functions provided to determine their period, amplitude, range, and domain.

### **First Function (Green Graph)**

1. **Period:**
   - The period of a function is the length of one complete cycle on the x-axis. From the graph, it seems that one full cycle repeats between \( -\pi \) and \( \pi \). 
   - **Period:** \( T = 2\pi \)

2. **Amplitude:**
   - The amplitude is the maximum absolute value the function reaches from its midline (usually the x-axis for simple sine or cosine functions).
   - From the graph, the maximum value is 1 and the minimum value is -1.
   - **Amplitude:** \( A = 1 \)

3. **Range:**
   - The range is the set of all possible output values (y-values).
   - Since the function oscillates between -1 and 1:
   - **Range:** \( [-1, 1] \)

4. **Domain:**
   - The domain is the set of all possible input values (x-values) for the function.
   - Since this is a sine or cosine function, it is defined for all real numbers.
   - **Domain:** \( (-\infty, \infty) \)

### **Second Function (Red Graph)**

1. **Period:**
   - This graph appears to be a function like \( y = \sin(kx) \) where \( k \) affects the period. One full cycle repeats every \( \frac{2\pi}{5} \) units.
   - **Period:** \( T = \frac{2\pi}{5} \)

2. **Amplitude:**
   - The amplitude is the maximum distance from the midline (x-axis) to the peak of the wave. The graph oscillates between 1 and -1.
   - **Amplitude:** \( A = 1 \)

3. **Range:**
   - The function oscillates between -1 and 1.
   - **Range:** \( [-1, 1] \)

4. **Domain:**
   - This sine function is defined for all real numbers.
   - **Domain:** \( (-\infty, \infty) \)

Would you like more details on any of these properties, or do you have any questions?

### Related Questions:
1. How does the value of \( k \) in \( y = \sin(kx) \) affect the period of the function?
2. What would happen to the amplitude if the equation was \( y = 2\sin(x) \)?
3. How do you determine the phase shift of a sine or cosine function from its graph?
4. How do you find the vertical shift in a sine or cosine function?
5. How does the period of a function change if the function is \( y = \sin(\frac{1}{2}x) \)?

### Tip:
Remember that the amplitude of a sine or cosine function is always the absolute value of the coefficient of the sine or cosine term in the function's equation.

Explore the analysis of periodic functions to determine their period, amplitude, range, and domain. This problem covers trigonometric concepts including the period of sine and cosine functions. Suitable for high school students.

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