Glossary
Amplitude (a)
The maximum displacement or height of a wave from its midline, always expressed as a positive value.
Example:
A large amplitude in an ocean wave indicates a very tall wave, potentially dangerous for small boats.
Cosine Function (General Equation)
The standard algebraic form $f( heta) = a\cos(b(\theta + c)) + d$ used to represent sinusoidal waves, often preferred when the wave starts at a maximum or minimum value.
Example:
If a wave begins its cycle at its highest point at , a cosine function is often the most natural choice for its equation.
Midline
The horizontal line that passes exactly halfway between the maximum and minimum values of a sinusoidal function, determined by the vertical translation (d).
Example:
For a wave oscillating between a high of 10 and a low of 2, the midline would be at y=6.
Period (T)
The horizontal length of one complete cycle of a sinusoidal wave, representing the interval over which the wave's pattern repeats.
Example:
The time it takes for a planet to complete one orbit around the sun is its orbital period.
Phase Shift (c)
The horizontal translation of a sinusoidal wave, indicating how much the wave has moved left or right from its standard starting position.
Example:
If a sound wave's peak arrives later than expected, it has experienced a positive phase shift.
Sine Function (General Equation)
The standard algebraic form $f( heta) = a\sin(b(\theta + c)) + d$ used to represent and analyze sinusoidal waves.
Example:
To describe the height of a point on a rotating Ferris wheel over time, you would typically use a sine function equation.
Sinusoidal Functions
Wave-like graphs based on the sine and cosine functions, characterized by their repeating, periodic nature.
Example:
The daily temperature fluctuations throughout a year can often be modeled effectively using sinusoidal functions.
Vertical Translation (d)
The vertical shift of a sinusoidal wave, which also defines the horizontal line known as the midline of the function.
Example:
Raising the entire graph of a wave up by 5 units would result in a vertical translation of +5.