Glossary
Amplitude
For a periodic function like sine or cosine, the amplitude is half the distance between the maximum and minimum values of the function, representing the vertical stretch from the midline.
Example:
In the function y = 4sin(x), the Amplitude is 4, meaning the wave extends 4 units above and below its midline.
Cosine
A trigonometric function that, for an angle in standard position on the unit circle, is the x-coordinate of the point where the angle's terminal side intersects the circle. In a right triangle, it's the ratio of the adjacent side to the hypotenuse.
Example:
The Cosine of π radians is -1, indicating the x-coordinate at that position on the unit circle.
Maximum Value
The highest point or greatest output value that a function reaches within its domain or a specified interval.
Example:
For the function f(x) = sin(x), the Maximum Value is 1.
Minimum Value
The lowest point or smallest output value that a function reaches within its domain or a specified interval.
Example:
For the function f(x) = cos(x), the Minimum Value is -1.
Oscillation
The repetitive variation or fluctuation of a value or quantity, typically between two extreme points. In graphs, it describes the wave-like movement of functions like sine and cosine.
Example:
The up-and-down movement of a spring with a weight attached demonstrates Oscillation.
Periodicity
The property of a function where its values repeat over a regular, fixed interval. For standard sine and cosine functions, this interval is 2π radians.
Example:
The repeating pattern of seasons throughout the year is an example of Periodicity in nature.
Sine
A trigonometric function that, for an angle in standard position on the unit circle, is the y-coordinate of the point where the angle's terminal side intersects the circle. In a right triangle, it's the ratio of the opposite side to the hypotenuse.
Example:
The Sine of π/2 radians is 1, representing the y-coordinate at the top of the unit circle.
Transformations
Changes applied to the graph of a parent function, including shifts (horizontal or vertical), stretches or compressions, and reflections, which alter its position, size, or orientation.
Example:
Applying a horizontal shift and a vertical stretch to y = cos(x) are examples of graph Transformations.
Unit Circle
A circle with a radius of 1 unit, centered at the origin (0,0) of a coordinate plane, used to define trigonometric functions for any angle.
Example:
To find the sine and cosine of π/4, we locate the corresponding point on the Unit Circle and read its coordinates.
Vertical Shift
A transformation that moves the entire graph of a function up or down by a constant amount, determined by adding or subtracting a value from the function's output.
Example:
The function y = cos(x) + 3 has a Vertical Shift of 3 units upwards compared to y = cos(x).
X-intercept
The point(s) where the graph of a function crosses or touches the x-axis, meaning the y-coordinate (or function output) is zero.
Example:
The point (π, 0) is an X-intercept for the graph of y = sin(x).