2.02: Graphing Trigonometric Functions

Odd Function

A function in which f(-x) = -f(x) for all values of x; the graph is symmetrical about the origin.

Even Function

A function in which f(-x) = f(x) for all values of x; the graph is symmetrical about the y-axis.

Periodic Function

A function that repeats it values over regular intervals.

Phase Shift

A horizontal displacement, or translation, of a periodic function.

Midline

A horizontal line that is equidistant between the maximum and minimum values of a periodic curve.

Sinusoid

A sine or cosine curve that is a periodic wave.

Cofunctions

A trigonometric function whose value for the complement of an angle is equal to the value of a given trignonometric function of the angle itself; sine is the cofunction of cosine, and cosine is the cofunction of sine.

Key Features of the Tangent Functions

Domain: The domain of the tangent function is almost all real numbers, but it excludes the undefined parts when x (cosine) is equal to zero, or cos x= 0. The cosine functions is equal to pi/2 and 3pi/2. Can extend this pattern to list the domain as being equal to all real numbers expect where x = pi/2 +npi, where n is any integer (positive or negative). Period: The function repeats its values, these regions serve as literal starting and ending points for each period of the function. The period of the tangent function is pi. Range: Range of the tangent function is equal to all real numbers.

Amplitude

Half of the difference between the maximum and minimum values of a periodic curve.

Tangent Functions

Has a different domain and range from the sine and cosine functions. It isn't continuous everywhere. The key features that describe the sinusoidal functions of sine and cosine don't all apply to the tangent function since there are several places where the tangent function is undefined. Amplitude of a tangent function doesn't make since because it doesn't have maximum and minimum values. There is not midline because there isn't a amplitude, which you need to have to get a midline.

Period

The horizontal distance that a function takes to complete one full cycle.

f(x) = A sin(Bx+C) +D

f(x): the sinusoidal function value, or the output. A: Gives the amplitude of the function. Amplitude is always a positive magnitude, |A|. If A is negative, the curve is vertically flipped across it's midline. B: Helps us find the period of the function. The period of a sinusoidal function is always determined by this formula: period = 2pi/B C: find the horizontal translation of the sinusoidal function, also called the phase shift, and its found with this formula: phase shift: C/B. If the ratio of C/B is positive, the shift is to the left, and if the ratio is negative, the shift is to the right.