Ask a New Question
Menu

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.

 

 

 

1 answer

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.