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:
1. Seasons: The yearly cycle of seasons (spring, summer, fall, winter) occurs due to the Earth's axial tilt and orbit around the Sun.
2. Tides: Ocean tides are periodic motions caused by the gravitational pull of the Moon and the Sun.
3. Day and Night: The rotation of the Earth results in the daily cycle of day and night.
4. Heartbeats: The rhythmic contraction of heart muscles creates a periodic pattern in the heart rate.
5. Sound Waves: The vibrations in a medium that produce sound are periodic, characterized by frequency and wavelength.
6. 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
1. Amplitude: The coefficient (2) in front of the sine function indicates the amplitude, the maximum displacement from the midline. Here, the amplitude is 2.
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.
3. 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.
4. Phase Shift: There is no horizontal shift in this function since there are no added or subtracted terms with x.
5. 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
1. Amplitude: The amplitude is 5, meaning the graph will rise or fall 5 units from its midline.
2. Midline: The midline is given by -2, placing the midpoint of the oscillation at y=-2.
3. Period: For the cosine function, the period is 2pi/b; here, b=6pi, so the period is 2pi/6pi=1/3.
4. Graph Behavior: The cosine function oscillates between the midline ± amplitude. Hence, G(x) oscillates between -2-5=-7 and -2+5=3.
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1. Seasons: The yearly cycle of seasons (spring, summer, fall, winter) occurs due to the Earth's axial tilt and orbit around the Sun.
2. Tides: Ocean tides are periodic motions caused by the gravitational pull of the Moon and the Sun.
3. Day and Night: The rotation of the Earth results in the daily cycle of day and night.
4. Heartbeats: The rhythmic contraction of heart muscles creates a periodic pattern in the heart rate.
5. Sound Waves: The vibrations in a medium that produce sound are periodic, characterized by frequency and wavelength.
6. 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
1. Amplitude: The coefficient (2) in front of the sine function indicates the amplitude, the maximum displacement from the midline. Here, the amplitude is 2.
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.
3. 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.
4. Phase Shift: There is no horizontal shift in this function since there are no added or subtracted terms with x.
5. 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
1. Amplitude: The amplitude is 5, meaning the graph will rise or fall 5 units from its midline.
2. Midline: The midline is given by -2, placing the midpoint of the oscillation at y=-2.
3. Period: For the cosine function, the period is 2pi/b; here, b=6pi, so the period is 2pi/6pi=1/3.
4. Graph Behavior: The cosine function oscillates between the midline ± amplitude. Hence, G(x) oscillates between -2-5=-7 and -2+5=3.
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Answers
GPT-4o mini
Periodic phenomena are events that repeat at regular intervals. Here are some examples:
1. **Seasons**: Yearly cycle due to Earth's tilt and orbit around the Sun.
2. **Tides**: Ocean tides caused by the gravitational pull of the Moon and Sun.
3. **Day and Night**: Daily cycle resulting from Earth's rotation.
4. **Heartbeats**: Rhythmic contractions of heart muscles.
5. **Sound Waves**: Periodic vibrations in a medium.
6. **Pendulum Motion**: Back-and-forth swinging motion.
### Analysis of Functions
**1. \(f(x) = 2 \sin(3x) + 5\)**
- **Amplitude**: 2
- **Midline**: \(y = 5\)
- **Period**: \(\frac{2\pi}{3}\)
- **Phase Shift**: None
- **Graph Behavior**: Oscillates between 3 and 7.
**2. \(g(x) = 5 \cos(\pi x) - 2\)**
- **Amplitude**: 5
- **Midline**: \(y = -2\)
- **Period**: \(\frac{2}{\pi}\)
- **Graph Behavior**: Oscillates between -7 and 3.
1. **Seasons**: Yearly cycle due to Earth's tilt and orbit around the Sun.
2. **Tides**: Ocean tides caused by the gravitational pull of the Moon and Sun.
3. **Day and Night**: Daily cycle resulting from Earth's rotation.
4. **Heartbeats**: Rhythmic contractions of heart muscles.
5. **Sound Waves**: Periodic vibrations in a medium.
6. **Pendulum Motion**: Back-and-forth swinging motion.
### Analysis of Functions
**1. \(f(x) = 2 \sin(3x) + 5\)**
- **Amplitude**: 2
- **Midline**: \(y = 5\)
- **Period**: \(\frac{2\pi}{3}\)
- **Phase Shift**: None
- **Graph Behavior**: Oscillates between 3 and 7.
**2. \(g(x) = 5 \cos(\pi x) - 2\)**
- **Amplitude**: 5
- **Midline**: \(y = -2\)
- **Period**: \(\frac{2}{\pi}\)
- **Graph Behavior**: Oscillates between -7 and 3.