Glossary
Amplitude
The vertical distance from the midline to either the maximum or minimum value of a sinusoidal function, always expressed as a positive value.
Example:
A larger amplitude in a spring's oscillation means the spring stretches and compresses further from its resting position.
Concave Down
A characteristic of a curve where it opens downwards, resembling a 'frown' or an inverted cup.
Example:
At the peak of a hill on a roller coaster, the track is concave down, making riders feel lighter.
Concave Up
A characteristic of a curve where it opens upwards, resembling a 'smile' or a cup that can hold water.
Example:
On a roller coaster, the track is concave up at the bottom of a dip, pushing riders into their seats.
Even Symmetry
A property of a function where its graph is symmetric about the y-axis, meaning that if you reflect the graph over the y-axis, it remains unchanged. Mathematically, f(-x) = f(x).
Example:
The cosine function, cos(x), demonstrates even symmetry because cos(-x) = cos(x).
Frequency (f)
The number of complete cycles that occur in one unit of time, calculated as the reciprocal of the period.
Example:
A sound wave with a high frequency means more cycles per second, resulting in a higher-pitched sound.
Midline (k)
The horizontal line that passes through the center of a sinusoidal wave, representing the average of the function's maximum and minimum y-values.
Example:
For a tide chart showing water depth, the midline would represent the average water level over time, halfway between high and low tide.
Odd Symmetry
A property of a function where its graph is symmetric about the origin, meaning that if you rotate the graph 180 degrees, it remains unchanged. Mathematically, f(-x) = -f(x).
Example:
The sine function, sin(x), exhibits odd symmetry because sin(-x) = -sin(x).
Oscillation
The repetitive variation, typically in time, of some measure about a central value or between two or more different states.
Example:
The back-and-forth movement of a pendulum is a classic example of oscillation.
Period (T)
The horizontal length of one complete cycle of a periodic function, representing the distance before the wave pattern begins to repeat itself.
Example:
If a Ferris wheel completes a full rotation in 2 minutes, its motion can be modeled by a sinusoidal function with a period of 2 minutes.
Sinusoidal function
A function that exhibits a wave-like, oscillating, and periodic pattern, resembling a sine or cosine curve. Both sine and cosine functions, along with their transformations, are examples of sinusoidal functions.
Example:
The daily temperature in a city often follows a sinusoidal function throughout the year, peaking in summer and dipping in winter.