2 A 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
3 Graph 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
4 Graph 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
5 The 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)
6 We 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
7 Now 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!!
8 On 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!!!