Glossary

Amplitude (a)

Criticality: 3

Half the distance between the maximum and minimum values of a sinusoidal function, representing the vertical stretch from the midline.

Example:

If a Ferris wheel's height ranges from 2 feet to 50 feet, its amplitude is 24 feet, indicating how far it moves from its center height.

Cosine function

Criticality: 2

A trigonometric function that describes a smooth, oscillating wave, typically starting at its maximum value.

Example:

The horizontal position of a point on a rotating wheel, starting from its rightmost point, can be modeled by a cosine function.

Frequency (b)

Criticality: 3

A parameter in sinusoidal equations that determines how many cycles occur within a standard interval, inversely related to the period.

Example:

In the equation $f(x) = \sin(2x)$ , the frequency of 2 means the wave completes two full cycles in the interval of $2\pi$ .

Horizontal shift (c)

Criticality: 3

The horizontal displacement of a sinusoidal graph from its standard position, indicating how much the wave is shifted left or right.

Example:

If a sound wave's peak is delayed by 0.5 seconds, its equation would include a horizontal shift of -0.5 (or +0.5 depending on the form $x-c$ or $x+c$ ).

Midline

Criticality: 2

The horizontal line that passes exactly halfway between the maximum and minimum values of a sinusoidal function, representing its vertical center.

Example:

For a wave oscillating between a height of 10 feet and 2 feet, the midline would be at 6 feet, serving as the wave's equilibrium position.

Period

Criticality: 3

The horizontal length of one complete cycle of a sinusoidal function before the pattern repeats.

Example:

If a pendulum swings back and forth and returns to its starting point every 3 seconds, its period is 3 seconds.

Phase shift

Criticality: 3

Another term for horizontal shift, describing how much a sinusoidal graph is moved left or right from its usual starting point.

Example:

If a sine wave normally starts at (0,0) but now starts at ( $\pi/2$ ,0), it has a phase shift of $\pi/2$ to the right.

Sine function

Criticality: 2

A trigonometric function that describes a smooth, oscillating wave, typically starting at its midline and increasing.

Example:

The height of a point on a rotating wheel, starting from the horizontal axis and moving upwards, can be modeled by a sine function.

Sinusoidal functions

Criticality: 3

Functions whose graphs are smooth, repeating waves, typically represented by sine or cosine equations.

Example:

The daily temperature in a city over a year can be modeled by a sinusoidal function, showing a repeating pattern of highs and lows.

Transformations

Criticality: 2

Changes applied to a parent function (like sine or cosine) that alter its graph by stretching, compressing, shifting, or reflecting it.

Example:

Changing the 'a' value in $f(x) = a\sin(x)$ causes a vertical stretch or compression, which is a type of transformation.

Vertical shift (d)

Criticality: 3

The vertical displacement of the midline of a sinusoidal function from the x-axis, representing the average value of the function.

Example:

If the average water level in a harbor is 5 meters, the vertical shift of the tide function would be 5.