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Can you think of other real-world examples of periodic phenomena? You have learned about using the unit circle to evaluate the sine and cosine functions as well as some of the patterns that naturally occur. What if you were asked to graph functions like f(x)=2sin(3x)+5 or g(x)=5cos(6πx)−2 ? What would these graphs look like?

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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 in nature, 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 \( g(x) = 5\cos(6\pi x)−2 \).

Graph of \( f(x) = 2\sin(3x) + 5 \)

  1. Amplitude: The coefficient (2) in front of the sine function indicates the amplitude, which is 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 in this case. 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 \( \frac{2\pi}{b} \), where \( b \) is the coefficient in front of \( x \). For \( f(x) \), \( b = 3 \), so the period is \( \frac{2\pi}{3} \).
  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 \( 5 + 2 = 7 \).

Graph of \( g(x) = 5\cos(6\pi x)−2 \)

  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 \( \frac{2\pi}{b} \). Here, \( b = 6\pi \), so the period is \( \frac{2\pi}{6\pi} = \frac{1}{3} \).
  4. Phase Shift: Like the sine function, there is no horizontal shift here.
  5. Graph Behavior: The cosine function oscillates between the midline ± amplitude. Hence, \( g(x) \) oscillates between \( -2 - 5 = -7 \) and \( -2 + 5 = 3 \).

Summary of the Graphs

  1. Graph of \( f(x) = 2\sin(3x)+5 \):

    • Midline at \( y = 5 \)
    • Oscillates between \( 3 \) and \( 7 \)
    • Period of \( \frac{2\pi}{3} \)

  2. Graph of \( g(x) = 5\cos(6\pi x)−2 \):

    • Midline at \( y = -2 \)
    • Oscillates between \( -7 \) and \( 3 \)
    • Period of \( \frac{1}{3} \)

These graphs can be plotted on a coordinate plane to visualize their periodic nature and how they oscillate around their respective midlines.