Sinusoidal Functions

Sinusoidal functions describe smooth, periodic oscillations and are used extensively in mathematics, physics, engineering, and signal processing. The most well-known sinusoidal functions are based on the sine and cosine functions.

Sinusoidal Graph

A sinusoidal graph refers to a curve resembling the graph of the sine function or the cosine function. These graphs exhibit periodic behavior, meaning their pattern repeats over regular intervals.
  • Domain: All real numbers, .
  • Range: .
  • Period: (the pattern of the graph repeats in intervals of ).
  • Amplitude: (the distance from the midline, , to the maximum or minimum points).
  • Zeros: , where is an integer (the sine graph has zeros at every integer multiple of ).
  • Odd Function: , meaning the graph is symmetric about the origin.
  • Domain: All real numbers, .
  • Range: .
  • Period: (the pattern of the graph repeats in intervals of ).
  • Amplitude: (the distance from the midline, , to the maximum or minimum points).
  • Zeros: , where is an integer (the cosine graph has zeros at every odd multiple of ).
  • Even Function: , meaning the graph is symmetric about the -axis.

While all sinusoidal graphs share the general shape of and , their specific characteristics (like amplitude, period, and range) can vary significantly depending on the equation.

General Formula for Sinusoidal Functions

The generalized form of a sinusoidal function is:
  1. Amplitude
    The amplitude represents half the distance between the maximum and minimum values of the graph. It is given by .
  2. Period
    The period is the horizontal length of one complete cycle of the graph, given by .
  3. Frequency
    The frequency is the reciprocal of the period, representing the number of cycles the graph completes in a unit interval. It is calculated as .
  4. Midline
    The graph is centered around the horizontal line , called the midline.
  5. Phase Shift
    The graph shifts horizontally by . A positive shifts the graph to the right, and a negative shifts it to the left.
  6. Vertical Shift
    The graph moves vertically by . A positive shifts the graph upward, and a negative shifts it downward.
  7. Reflection
    Reflection occurs when the graph is flipped across the -axis or -axis. If , the graph is reflected across the -axis.

Example: Graphing Sinusoidal Functions

Graph :
  • Amplitude:
  • Period:
  • Horizontal Shift: , so the graph shifts right
  • Vertical Shift: , so the graph shifts up
  • Midline:

Writing the Equation of a Sinusoidal Graph

To determine the equation of a sinusoidal function from its graph:
  1. Midline (Find ):
    The midline is the average of the maximum and minimum values.
  2. Amplitude (Find ):
    The amplitude is the distance from the midline to a maximum or minimum.
  3. Period (Find ):
    Measure the horizontal distance between two consecutive peaks or troughs. The period is .
  4. Phase Shift (Find ):
    Determine the horizontal shift by finding where the sinusoid first intersects the midline before reaching a maximum.

Example: Write the Equation

Given a sinusoidal graph:
  • Maximum:
  • Minimum:
  • Midline:
  • Amplitude:
  • Period: , so
  • Phase Shift: The graph shifts horizontally by units.
The equation is: