3.2 Modeling data and scenarios with sinusoidal functions

• Sinusoidal functions are trigonometric functions that oscillate in a repeating pattern
• The amplitude of a sinusoidal function represents the vertical distance from the midline to the maximum or minimum value
• The vertical shift of a sinusoidal function is represented by the variable $D$


  True
• A positive phase shift translates the graph to the left

  False
• Match the characteristic with its transformation:

  Amplitude ↔️ Vertical stretch or compression

  Period ↔️ Horizontal stretch or compression

  Phase Shift ↔️ Horizontal translation

  Vertical Shift ↔️ Vertical translation
• How is the period of a sinusoidal function affected by the coefficient $B$?


  Horizontal stretch or compression
• What type of transformation does the period of a sinusoidal function represent?

  Horizontal stretch or compression
• The period of ocean tides is approximately 12.4 hours
• The period of a sinusoidal function is calculated using the coefficient $B$ in the equation


  True
• What is a phase shift in a sinusoidal function?

  Horizontal translation
• The period of a sinusoidal function is the length of one complete cycle
• The period of solar power generation is 24 hours.


  True
• What does the vertical shift parameter (D) in a sinusoidal function represent?

  Vertical translation
• Sinusoidal functions can model ocean tides and solar power generation due to their periodic behavior.

  True
• The amplitude of ocean tides represents the vertical distance from the average sea level to the high or low tide
• What does the amplitude $|A|$ represent in a sinusoidal function?


  Vertical distance from midline
• What does the vertical shift $D$ represent in a sinusoidal function?


  Upward or downward translation
• Match the characteristic with its definition in the general form of a sinusoidal function:

  Amplitude ↔️ The vertical distance from the midline to the maximum or minimum value

  Period ↔️ The length of one complete cycle of the function

  Phase Shift ↔️ The horizontal shift of the graph

  Vertical Shift ↔️ The upward or downward translation of the graph
• The amplitude of a sinusoidal function represents the vertical distance from the midline to the maximum or minimum value
• Match the transformation with its effect on the standard sine or cosine function:

  Phase Shift ↔️ Horizontal translation

  Vertical Shift ↔️ Vertical translation

  Period ↔️ Horizontal stretch or compression
• Match the characteristic with its real-world application:

  Amplitude ↔️ Variation in solar panel power efficiency

  Period ↔️ Approximately 12.4 hours for semidiurnal tide

  Phase Shift ↔️ Depends on sunrise and sunset times

  Vertical Shift ↔️ Average power output
• How is the period of a sinusoidal function calculated?

  $\frac{2\pi}{|B|}$
• What does the phase shift of a sinusoidal function represent?

  The horizontal shift
• The amplitude of a sinusoidal function is denoted by the symbol |A|
• The period of a sinusoidal equation is calculated as \frac{2\pi}{|B|}</latex> using the parameter B
• The first step in using a sinusoidal equation for problem-solving is to identify the parameters
• What does the period of a sinusoidal function represent?

  Length of one complete cycle
• How does the amplitude affect the graph of a sinusoidal function?

  Vertical stretch or compression
• A vertical shift of $D$ moves the graph up or down
• Sinusoidal functions can model phenomena like tides and sound waves


  True
• The phase shift $C$ in a sinusoidal equation represents a horizontal translation of the graph


  True
• The key characteristics of sinusoidal functions include amplitude, period, phase shift, and vertical shift


  True
• The phase shift of a sinusoidal function corresponds to a horizontal translation of the graph


  True
• The vertical shift in a sinusoidal model of solar power generation represents the average power output

  True
• Which component of a sinusoidal equation represents the vertical translation of the graph?

  D</latex>
• A vertical shift of a sinusoidal function results in a horizontal translation.

  False
• Match the characteristic with its definition:

  Amplitude ↔️ Vertical distance from midline

  Period ↔️ Length of one complete cycle

  Phase Shift ↔️ Horizontal shift

  Vertical Shift ↔️ Vertical translation
• What are the two general forms of sinusoidal functions?

  $y =$$A \sin(B(x - C)) +$$D$ and $y =$$A \cos(B(x - C)) +$$D$
• Sinusoidal functions can model ocean tides and solar power generation by adjusting their parameters.

  True
• Match the characteristic with its example in ocean tides and solar power generation:

  Amplitude ↔️ Vertical distance from average sea level to high/low tide

  Period ↔️ Approximately 12.4 hours (semidiurnal tide)

  Phase Shift ↔️ Depends on location and tidal patterns

  Vertical Shift ↔️ Average sea level