Pre-AP Pre-Calculus Chapter 4, Section 4 Graphs of Sine and Cosine: Sinusoids 2013 - 2014

Basic Function: The Sine Function Window: [-2π, 2π] by [-4, 4] Domain: Range: Continuity: Inc/Dec: Asymptotes: End Behavior: Period:

Basic Function: The Cosine Function Window: [-2π, 2π] by [-4, 4] Domain: Range: Continuity: Inc/Dec: Asymptotes: End Behavior: Period:

Sinusoids & Transformations Sinusoid: A function is a sinusoid if it can be written in the form ____________________ where a, b, c, and d are constants and neither a nor b is 0. Amplitude: The amplitude of the sinusoid ______________________ 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.

Vertical Stretch or Shrink and Amplitude Find the amplitude of each function and use the language of transformations to describe how the graphs are related. Sketch each graph. 𝑦= cos 𝑥 𝑦= 1 2 cos 𝑥 𝑦=−3 cos 𝑥

Transformations Describe the transformation required to obtain 𝑦 2 from 𝑦 1 . 𝑦 1 = cos 2𝑥 𝑦 2 = 5 3 cos 2𝑥

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

Horizontal Stretch or Shrink and Period Find the period of each function and use the language of transformations to describe how the graphs are related. Sketch the graphs. 𝑦= sin 𝑥 𝑦=−2 sin 𝑥 3 𝑦=3 sin (−2𝑥)

Frequency of a Sinusoid The frequency of the sinusoid ______________________ is 𝑏 /2𝜋. The same can be said for the cosine version of the function. Graphically, the frequency is the number of complete cycles the wave complete in a unit interval. Hopefully you noticed that the frequency is the reciprocal of the _________.

Finding the Frequency of a Sinusoid Find the frequency of the function 𝑓 𝑥 =4 sin 2𝑥/3 and interpret its meaning graphically. Sketch the graph in the window [-3π, 3π] by [-4, 4].

Graphs of Sinusoids The graphs of 𝑦=𝑎 sin 𝑏 𝑥−ℎ +𝑘 and 𝑦=𝑎 cos 𝑏 𝑥−ℎ +𝑘 (where a nor b = 0) have the following characteristics: 𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒= 𝑎 𝑝𝑒𝑟𝑖𝑜𝑑= 2𝜋 𝑏 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦= 𝑏 2𝜋 When compared to the graphs of 𝑦=𝑎 sin 𝑏𝑥 and 𝑦=𝑎 cos 𝑏𝑥 , respectively, they also have the following characteristics: A phase shift of h A vertical translation of k

Constructing a Sinusoid by Transformations Construct a sinusoid 𝑦=𝑓(𝑥) that rises from a minimum value 𝑦=5 and 𝑥=0 to a maximum value of 𝑦=25 at 𝑥=32.

Graph one period of the function NOT using the calculator. 𝑦=3 cos 𝑥

Graph 3 periods of the function NOT using the calculator

Ch. 4.4 Homework Pg. 392 – 394, #’s: 4, 9, 14, 16, 17, 28, 29, 36, 37, 53, 58, 62 12 total problems Gray book: pg. 357 - 358