Question
Certainly! Periodic phenomena are events or quantities that repeat at regular intervals, and they can be observed in various aspects of nature and daily life. Here are some real-world examples of periodic phenomena:
Seasons: The yearly cycle of seasons (spring, summer, fall, winter) occurs due to the Earth's axial tilt and orbit around the Sun.
Tides: Ocean tides are periodic motions caused by the gravitational pull of the Moon and the Sun.
Day and Night: The rotation of the Earth results in the daily cycle of day and night.
Heartbeats: The rhythmic contraction of heart muscles creates a periodic pattern in the heart rate.
Sound Waves: The vibrations in a medium that produce sound are periodic, characterized by frequency and wavelength.
Pendulum Motion: A swinging pendulum exhibits periodic motion as it moves back and forth.
Now let's analyze the functions f(x)=2 sin(3x) +5 and g(x)=5 cos(pix)-2.
Graph of functions f(x)=2 sin(3x) + 5
Amplitude: The coefficient (2) in front of the sine function indicates the amplitude, the maximum displacement from the midline. Here, the amplitude is 2.
Midline: The midline is determined by the constant added to the function, which is 5. So, the midline of the graph is the horizontal line y=5.
Period: The period of a sine function is calculated using the formula 2pi/b, where b is the coefficient in front of x for f(x); b=3 so the period is 2pi/3.
Phase Shift: There is no horizontal shift in this function since there are no added or subtracted terms with x.
Graph Behavior: The sine function oscillates between the midline ± amplitude. Therefore, f(x) oscillates between 5-2=3 and 5+7=7.
Graph of g(x) = 5 cos (pi x) - 2
Amplitude: The amplitude is 5, meaning the graph will rise or fall 5 units from its midline.
Midline: The midline is given by -2, placing the midpoint of the oscillation at y=-2.
Period: For the cosine function, the period is 2pi/b; here, b=6pi, so the period is 2pi/6pi=1/3.
Graph Behavior: The cosine function oscillates between the midline ± amplitude. Hence, G(x) oscillates between -2-5=-7 and -2+5=3.
Seasons: The yearly cycle of seasons (spring, summer, fall, winter) occurs due to the Earth's axial tilt and orbit around the Sun.
Tides: Ocean tides are periodic motions caused by the gravitational pull of the Moon and the Sun.
Day and Night: The rotation of the Earth results in the daily cycle of day and night.
Heartbeats: The rhythmic contraction of heart muscles creates a periodic pattern in the heart rate.
Sound Waves: The vibrations in a medium that produce sound are periodic, characterized by frequency and wavelength.
Pendulum Motion: A swinging pendulum exhibits periodic motion as it moves back and forth.
Now let's analyze the functions f(x)=2 sin(3x) +5 and g(x)=5 cos(pix)-2.
Graph of functions f(x)=2 sin(3x) + 5
Amplitude: The coefficient (2) in front of the sine function indicates the amplitude, the maximum displacement from the midline. Here, the amplitude is 2.
Midline: The midline is determined by the constant added to the function, which is 5. So, the midline of the graph is the horizontal line y=5.
Period: The period of a sine function is calculated using the formula 2pi/b, where b is the coefficient in front of x for f(x); b=3 so the period is 2pi/3.
Phase Shift: There is no horizontal shift in this function since there are no added or subtracted terms with x.
Graph Behavior: The sine function oscillates between the midline ± amplitude. Therefore, f(x) oscillates between 5-2=3 and 5+7=7.
Graph of g(x) = 5 cos (pi x) - 2
Amplitude: The amplitude is 5, meaning the graph will rise or fall 5 units from its midline.
Midline: The midline is given by -2, placing the midpoint of the oscillation at y=-2.
Period: For the cosine function, the period is 2pi/b; here, b=6pi, so the period is 2pi/6pi=1/3.
Graph Behavior: The cosine function oscillates between the midline ± amplitude. Hence, G(x) oscillates between -2-5=-7 and -2+5=3.
Answers
GPT-4o mini
Let's delve into the periodic phenomena and the analysis of the given functions \( f(x) \) and \( g(x) \) in detail.
### Real-World Examples of Periodic Phenomena:
1. **Seasons**: Due to Earth's axial tilt and orbit, we experience a repeating cycle of seasons.
2. **Tides**: Regular fluctuations in ocean water levels caused by the gravitational forces of the Moon and Sun, occurring approximately every 12.5 hours.
3. **Day and Night**: The Earth's rotation leads to a predictable cycle of light and darkness every 24 hours.
4. **Heartbeats**: The heart's rhythmic contractions produce a consistent pattern that can vary with activity, stress, and health.
5. **Sound Waves**: Sound is created by vibrations in a medium, displaying periodic characteristics defined by frequency and wavelength.
6. **Pendulum Motion**: A classic example of periodic motion, where a pendulum swings back and forth in a regular and predictable manner.
### Analysis of Functions
#### Function \( f(x) = 2\sin(3x) + 5 \)
1. **Amplitude**: The amplitude is given by the coefficient of the sine function, which is 2. This means the maximum displacement from the midline is 2 units.
2. **Midline**: The midline of the graph is shifted vertically due to the constant term added to the sine function. Here, the midline is set at \( y = 5 \).
3. **Period**: The period of a sine function can be calculated using the formula:
\[
\text{Period} = \frac{2\pi}{b}
\]
where \( b \) is the coefficient of \( x \). For \( f(x) \), \( b = 3 \):
\[
\text{Period} = \frac{2\pi}{3}
\]
4. **Phase Shift**: There is no horizontal shift or phase shift in this function since there are no added or subtracted terms involving \( x \).
5. **Graph Behavior**: The function oscillates above and below the midline:
\[
\text{Minimum} = 5 - 2 = 3, \quad \text{Maximum} = 5 + 2 = 7
\]
Thus, \( f(x) \) oscillates between 3 and 7.
#### Function \( g(x) = 5 \cos(\pi x) - 2 \)
1. **Amplitude**: The amplitude is 5, indicating that the graph will oscillate 5 units above and below the midline.
2. **Midline**: The midline here is at \( y = -2 \).
3. **Period**: For the cosine function, the formula is the same:
\[
\text{Period} = \frac{2\pi}{b}
\]
Here, we find \( b \):
\[
g(x) = 5\cos(\pi x) \Rightarrow b = \pi \quad \Rightarrow \quad \text{Period} = \frac{2\pi}{\pi} = 2
\]
4. **Graph Behavior**: This cosine function oscillates around the midline:
\[
\text{Minimum} = -2 - 5 = -7, \quad \text{Maximum} = -2 + 5 = 3
\]
Thus, \( g(x) \) oscillates between -7 and 3.
### Summary
- **Function \( f(x) \):**
- Amplitude: 2
- Midline: y = 5
- Period: \( \frac{2\pi}{3} \)
- Range: [3, 7]
- **Function \( g(x) \):**
- Amplitude: 5
- Midline: y = -2
- Period: 2
- Range: [-7, 3]
Both functions share periodic characteristics, highlighting the beauty of periodicity in mathematics and nature.
### Real-World Examples of Periodic Phenomena:
1. **Seasons**: Due to Earth's axial tilt and orbit, we experience a repeating cycle of seasons.
2. **Tides**: Regular fluctuations in ocean water levels caused by the gravitational forces of the Moon and Sun, occurring approximately every 12.5 hours.
3. **Day and Night**: The Earth's rotation leads to a predictable cycle of light and darkness every 24 hours.
4. **Heartbeats**: The heart's rhythmic contractions produce a consistent pattern that can vary with activity, stress, and health.
5. **Sound Waves**: Sound is created by vibrations in a medium, displaying periodic characteristics defined by frequency and wavelength.
6. **Pendulum Motion**: A classic example of periodic motion, where a pendulum swings back and forth in a regular and predictable manner.
### Analysis of Functions
#### Function \( f(x) = 2\sin(3x) + 5 \)
1. **Amplitude**: The amplitude is given by the coefficient of the sine function, which is 2. This means the maximum displacement from the midline is 2 units.
2. **Midline**: The midline of the graph is shifted vertically due to the constant term added to the sine function. Here, the midline is set at \( y = 5 \).
3. **Period**: The period of a sine function can be calculated using the formula:
\[
\text{Period} = \frac{2\pi}{b}
\]
where \( b \) is the coefficient of \( x \). For \( f(x) \), \( b = 3 \):
\[
\text{Period} = \frac{2\pi}{3}
\]
4. **Phase Shift**: There is no horizontal shift or phase shift in this function since there are no added or subtracted terms involving \( x \).
5. **Graph Behavior**: The function oscillates above and below the midline:
\[
\text{Minimum} = 5 - 2 = 3, \quad \text{Maximum} = 5 + 2 = 7
\]
Thus, \( f(x) \) oscillates between 3 and 7.
#### Function \( g(x) = 5 \cos(\pi x) - 2 \)
1. **Amplitude**: The amplitude is 5, indicating that the graph will oscillate 5 units above and below the midline.
2. **Midline**: The midline here is at \( y = -2 \).
3. **Period**: For the cosine function, the formula is the same:
\[
\text{Period} = \frac{2\pi}{b}
\]
Here, we find \( b \):
\[
g(x) = 5\cos(\pi x) \Rightarrow b = \pi \quad \Rightarrow \quad \text{Period} = \frac{2\pi}{\pi} = 2
\]
4. **Graph Behavior**: This cosine function oscillates around the midline:
\[
\text{Minimum} = -2 - 5 = -7, \quad \text{Maximum} = -2 + 5 = 3
\]
Thus, \( g(x) \) oscillates between -7 and 3.
### Summary
- **Function \( f(x) \):**
- Amplitude: 2
- Midline: y = 5
- Period: \( \frac{2\pi}{3} \)
- Range: [3, 7]
- **Function \( g(x) \):**
- Amplitude: 5
- Midline: y = -2
- Period: 2
- Range: [-7, 3]
Both functions share periodic characteristics, highlighting the beauty of periodicity in mathematics and nature.