2A function f is periodic if f (x + h) = f (x) for every x in domain of f Period of f = smallest positive number hOne cycle of graph is completed in each periodEx 1) Verify that the graph represents a periodic function and identify its period.Repeats in cycles periodic!Period = 4
3Graph of y = sin x Period = 2π x y1–1Period = 2πFor all x, sin (–x) = –sin x (odd function)Symmetric wrt originDomain = RRange = [–1, 1]Zeros occur at multiples of π
4Graph of y = cos x Period = 2π x y1–1Period = 2πFor all x, cos (–x) = cos x (even function)Symmetric wrt y-axisDomain = RRange = [–1, 1]Zeros occur at odd multiples of
5The sine and cosine functions are related to each other. They are called cofunctions.Ex 1) Express each function in terms of its cofunction.a)b)
6We will now take a look at how we can transform the basic sine & cosine curves Use Desmos app & the worksheet to help guide us.Open Desmos. Choose , then Trigonometry ,and then All the Trig FunctionsTap into box 7 and start deleting until all you are left with is box 2 sin (x)We would like to adjust the window so that the x-axis is showing [–2π, 2π] and the y-axis is [–5, 5]Pinch & spread with 2 fingers to get the window just right
7Now look at WS. A graph from [–2π, 2π] is pictured. We already know about parent graphs & transformations.Write down (and then share) what will happen to y = sin x if you graph #1 (and WHY).Now enter #1: ½ sin (x) in box 3(to get ½, simply type 1 ÷ 2)(to get sin, under tab and tab)Was your guess right?!Now, let’s repeat this process with #2 – 4.Please don’t go to back side until we are all ready!!
8On back of WS is y = cos x from [–2π, 2π]. We have just reminded ourselves & practiced graphing transformations using sine as the parent graph.Let’s see how quickly (and accurately) you can graph the 4 transformations of y = cos xOn Your Mark….All done! Quickly confirm with DesmosWe will do more involved transformations later in the chapter … today just the basics!Get Set….GO!!!