4.1 Graphs of the Sine and Cosine Functions

Compute the Period of a Function

• Period = 2π/B • Then divide this into 4 intervals: for example π would become 0, π/4, π/2, 3π/4, and π.

Amplitude

• The amplitude of a periodic function is one-half the absolute value of the difference between the maximum and minimum function values. It's always positive. • Describes the height of the graph both above and below a horizontal line passing through the "middle" of the graph.

Cosine Function f(x) = cos x

• The graph is continuous over its entire domain • Its x-intercepts are of the form (2n+1)π/2, where n is an integer. • Its period is 2π • The graph is symmetric with respect to the y-axis, so the function is an even function. For all x in the domain, cos(-x) = cos x

Sine Function f(x) = sin x

• The graph is continuous over its entire domain (-∞,∞) • Its x-intercepts are of the form nπ where n is an integer. • Its period is 2π • The graph is symmetric with respect to the origin, so the function is an odd function. For all x in the domain, sin(-x)=-sin x

Period of a Function

•The least possible positive value of p. • The number of radians that it takes to complete the cycle *The period of a basic sine and cosine function is 2π. The circumference of the unit circle is 2π, so the least possible value of p for which the sine and cosine functions repeat is 2π. • The graph of a function of the form y = sin bx or y = cos bx for b>0, will have a period different from 2π when b≠1.

Example: Graphing y = a sin bx (3 of 3)

*Note when a is negative, the graph of y = a sin bx is the reflection across the x-axis of the graph of y=|a| sin bx

Example: Graphing y = a cos bx for b that is a multiple of π (3 of 3)

*note when b is an integer multiple of π, the x-intercepts of the graph are rational numbers.

Periodic Function

A function whose graph repeats itself identically from left to right. represented by f(x) = f(x+np) for every real number x in the domain of f, every integer n, and some positive real number p.

Values of the Sine and Cosine Functions

A sine wave starts at 0 and increases to the maximum, decreases to the middle, continues to decrease to minimum, and then increases back to the middle ending the cycle. A cosine wave starts at its maximum, decreases to the middle, continues to decrease to its' minimum, then increases to the middle, continues to increase to the maximum ending the cycle.

Midline of a Periodic Function

the horizontal line that is of equal distance from both the maximum and the minimum points