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4.1 Graphs of the Sine & Cosine Functions. A function f is periodic if f ( x + h ) = f ( x ) for every x in domain of f Period of f = smallest positive number h One cycle of graph is completed in each period.
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A function f is periodic if f (x + h) = f (x) for every x in domain of f Period of f = smallest positive number h One cycle of graph is completed in each period Ex 1) Verify that the graph represents a periodic function and identify its period. Repeats in cycles periodic! Period = 4
1 x y Graph of y = sin x –1 • Period = 2π • For all x, sin (–x) = –sin x (odd function) • Symmetric wrt origin • Domain = R • Range = [–1, 1] • Zeros occur at multiples of π
x y 1 Graph of y = cosx –1 • Period = 2π • For all x, cos (–x) = cosx (even function) • Symmetric wrty-axis • Domain = R • Range = [–1, 1] • Zeros occur at odd multiples of
The sine and cosine functions are related to each other. They are called cofunctions. Ex 2) Express each function in terms of its cofunction. a) b)
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 Phase . Then delete all info in the boxes. 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
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 1 (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!!
On back of WS is y = cosx 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 = cosx On Your Mark…. All done! Quickly confirm with Desmos We will do more involved transformations later in the chapter … today just the basics! Get Set…. GO!!!
