5/7/13 Obj: SWBAT apply properties of periodic functions Bell Ringer: Construct a sinusoid with amplitude 2, period 3π, point 0,0 HW Requests: Pg 395 #72-75, 79, 80 WS Amplitude, Period, Phase Shift In class: Homework: Study for Quiz, Bring your Unit Circle Read Section 5.1 Project Due Wed. 5/8 Each group staple all projects together Education is Power! Dignity without compromise!

To find the phase or horizontal shift of a sinusoid Go to phase shift pdf

To find the phase or horizontal shift of a sinusoid Go to phase shift pdf

Horizontal Shift and Phase Shift (use Regent) Go to phase shift pdf

Determining the Period and Amplitude of y = a sin bx Given the function y = 3sin 4x, determine the period and the amplitude. The period of the function is Therefore, the period is.. The amplitude of the function is | a |. Therefore, the amplitude is 3. y = 3sin 4x

4.3.3 Graphing a Periodic Function Period: 2  Range: y-intercept: 0 x-intercepts: 0, ± , ±2 ,... Graph y = sin x. Amplitude: 1 1 Domain: all real numbers -1 ≤ y ≤ 1

4.3.4 Graphing a Periodic Function y-intercept: 1 x-intercepts:,... Period: 2  Domain: all real numbers Range: -1 ≤ y ≤ 1 Amplitude: 1 Graph y = cos x. 1

4.3.5 Graphing a Periodic Function Graph y = tan x. Asymptotes: Domain: Range: all real numbers Period: 

Determining the Period and Amplitude of y = a sin bx Sketch the graph of y = 2sin 2x. The period is  The amplitude is

Determining the Period and Amplitude of y = a sin bx Sketch the graph of y = 3sin 3x. The period is  The amplitude is

Writing the Equation of the Periodic Function Amplitude = 2 Period  b = 2 Therefore, the equation as a function of sine is y = 2sin 2x.

Writing the Equation of the Periodic Function Amplitude Period = 3 4  b = 0.5 Therefore, the equation as a function of cosine is y = 3cos 0.5x.

Summary of Transformations a = vertical stretch or shrink amplitude b = horizontal stretch or shrink period/frequency c = horizontal shift (phase shift) phase h = horizontal shift (phase shift) phase d = vertical translation/shift k = vertical translation/shift Exit Ticket pg 439 #61-64

Horizontal Shift and Phase Shift (use Regent)

Domain: Range: Continuity: Increasing/Decreasing Symmetry: Bounded: Max./Min. Horizontal Asymptotes Vertical Asymptotes End Behavior

Sinusoid – a function that can be written in the form below. Sine and Cosine are sinusoids. The applet linked below can help demonstrate how changes in these parameters affect the sinusoidal graph:

For each sinusoid answer the following questions. What is the midline? X = What is the amplitude? A = What is the period? T = (radians and degrees) What is the phase? =

Definition: A function y = f(t) is periodic if there is a positive number c such that f(t+c) = f(t) for all values of t in the domain of f. The smallest number c is called the period of the function. - a function whose value is repeated at constant intervals

Read page 388 – last paragraph Vertical Stretch and Shrink On your calculator baseline

Vertical Stretch and Shrink Amplitude of a graph Abs(max value – min value) 2 For graphing a sinusoid: To find the baseline or middle line on a graph y = max value – min value 2 Use amplitude to graph. baseline

Vertical Stretch and Shrink Amplitude of a graph Abs(max value – min value) 2 For graphing a sinusoid: To find the baseline or middle line on a graph y = max value – amplitude baseline

Horizontal Stretch and Shrink On your calculator Horizontal Stretch/Shrink y = f(cx) stretch if c< 1 factor = 1/c shrink if c > 1 factor = 1/c b = number complete cycles in 2π rad.

See if you can write the equation for the Ferris Wheel

We can use these values to modify the basic cosine or sine function in order to model our Ferris wheel situation.

28 Read page 388 – last paragraph Vertical Stretch and Shrink On your calculator

Horizontal Stretch and Shrink On your calculator Horizontal Stretch/Shrink y = f(bx) stretch if |b| < 1 shrink if |b |> 1 Both cases factor = 1/|b|

4.3.2 Periodic Functions Functions that repeat themselves over a particular interval of their domain are periodic functions. The interval is called the period of the function. In the interval there is one complete cycle of the function. To graph a periodic function such as sin x, use the exact values of the angles of 30 0, 45 0, and In particular, keep in mind the quadrantal angles of the unit circle. (1, 0)(-1, 0) (0, 1) (0, -1) The points on the unit circle are in the form (cosine, sine).

Determining the Amplitude of y = a sin x Graph y = 2sin x and y = 0.5sin x. y = sin x y = 2sin x y = sin x y = 0.5sin x 4.3.6

Period Amplitude Domain Range y = sin x y = 2sin xy = 0.5sin x 22 22 22 all real numbers -1 ≤ y ≤ 1-2 ≤ y ≤ ≤ y ≤ 0.5 Comparing the Graphs of y = a sin x The amplitude of the graph of y = a sin x is | a |. When a > 1, there is a vertical stretch by a factor of a. When 0 < a < 1, there is a vertical shrink by a factor of a

4.3.8 Determining the Period for y = sin bx, b > 0 y = sin x Graph y = sin 2x y = sin 2xy = sin x

Comparing the Graphs of y = sin bx Period Amplitude Domain Range y = sin x y = sin 2 xy = sin 0.5 x 22  44 111 all real numbers -1 ≤ y ≤ 1 The period for y = sin bx is When b > 1, there is a horizontal shrink. When 0 < b < 1, there is a horizontal stretch