Trigonometric Graphs (Precalculus)

Sinusoidal Function

A function that describes a continuous cycle. Sine and cosine are examples of sinusoidal functions. y = A sin (Bx - C) + D y = A cos (Bx - C) + D

Periodic Function

A function that returns the same output for different inputs at regular intervals. All trigonometric functions are periodic.

Phase Shift

A horizontal translation (left or right) of the graph of a trigonometric function. x = C/B

Asymptote

A line that a graph approaches, but never crosses. These lines are placed at input values that are undefined for the function.

Mid-line

A vertical translation (up or down) of the graph of a trigonometric function. Horizontal line equidistant from both the crest and the trough of a wave. y = D

Transformation

Changes made to the graph of a function. (Horizontal Shift, Vertical Shift, Dilation, Reflection)

Cosine Function

Function whose input (x) is an angle measure and whose output (y) is the horizontal leg of a right triangle. When plotted, the coordinates form a wave beginning at the crest or trough.

Secant Function

Function whose input (x) is an angle measure and whose output (y) is the ratio 1 over x.

Cosecant Function

Function whose input (x) is an angle measure and whose output (y) is the ratio of 1 over y.

Cotangent Function

Function whose input (x) is an angle measure and whose output (y) is the ratio x over y.

Tangent Function

Function whose input (x) is an angle measure and whose output (y) is the slope of a line.

Sine Function

Function whose input (x) is an angle measure and whose output (y) is the vertical leg of a right triangle. When plotted, the coordinates form a wave beginning at the node.

Undefined

Has no real solution in mathematics. Dividing by zero and taking the square root of negative numbers are two examples.

Period

Horizontal distance it takes a wave to complete one complete cycle. Period = 2π/B

Frequency

How many wavelengths there are over a certain time.

Node

Location where a sinusoidal graph crosses the midline. The locations on a wave where the amplitude is zero

Crest

Maximum output value (y) of a periodic function. Value of midline plus the amplitude.

Amplitude

Maximum vertical distance a wave varies (above or below) from its mid-line. Amplitude is always reported as positive. Amplitude = |A|

Trough

Minimum output value (y) of a periodic function. Value of the midline minus the amplitude.

Reciprocal

One of a pair of numbers whose product is 1. When the numerator and denominator of a fraction are inverted.

Trigonometric Ratio

Ratio that compares two sides of a right triangle for a given angle.

Limit

The output value (y) that a function "approaches" as the input (x) approaches some value from BOTH sides.

One Sided Limit

The output value (y) that a function "approaches" as the input (x) approaches some value from the right OR left side.

Domain

The set of all possible input values (x) for a function.

Range

The set of all possible output values (y) for a function.

x-intercept (zero, solution, root)

The value of the input (x) when the output (y) is zero. Where the graph of a function crosses the x-axis.

y-intercept

The value of the output (y) when the input (x) is zero. Where the graph of a function crosses the y-axis.

Evaluate a Function

To find the value of the output given an input, or visa-versa.