Chapter 4.4 Graphs of Sine and Cosine: Sinusoids Learning Target: Learning Target: I can generate the graphs of the sine and cosine functions along with the transformations of these graphs.

The Sine Function f(x) = sin x D: all real R: [-1,1] Continuous Inc and dec in periodic waves Odd: symmetric with respect to the origin Bounded Max: 1 Min: -1 No horizontal asymptotes No vertical asymptotes End Behavior: lim does not exist

The Cosine Function f(x) = cos x D: all real R: [-1,1] Continuous Inc and dec in periodic waves Even: symmetric with respect to the y-axis Bounded Max: 1 Min: -1 No horizontal asymptotes No vertical asymptotes End Behavior: lim does not exist

Definition: Sinusoid A function is a sinusoid if it can be written in the form f(x) = a sin (bx + c) + d where a, b, c, and d are constants and neither a nor b is 0.

Definition: Amplitude of a sinusoid The amplitude of the sinusoid f(x) = a sin (bx + c) + d is |a|. Similarly, the amplitude of f(x) = a cos (bx + c) + d is |a|. Graphically, the amplitude is half the height of the wave.

Example Find the amplitude of each function. A. y = cos x B. y = 1/2cos x C. y = -3 cos x

Period of a Sinusoid The period of the sinusoid f(x) = a sin (bx + c) + d is 2 П / |b|. This is the same for cosine. Graphically, the period is the length of one full cycle of the wave.

Frequency of a sinusoid The frequency of the sinusoid is |b|/2 П. Graphically the frequency is the number of complete cycles the wave completes in a unit interval.

Example Find the period of each function and use transformations to describe how the graphs are related. A. y = sin x B. y = -2sin (x/3) C. y = 3sin (-2x)

Homework Pg. 392 # 1 – 15 odd