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**4.4 Graphing by Addition of Ordinates**

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**The period of a sum function is found by examining the period of each individual part.**

The period is the first place the periods match up. For example, say 2 functions had periods of π and 2π. What’s the period of the sum function? list multiples of each until they match π 2π Something more challenging: Periods of 2π period = 2π period = π

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**Ex 1) Graph b = 1 c = 0 a = 2 d Per = IL: per = 2π per = π**

period = 2π Start by graphing (longer period) Check Period: b c a d = 1 = 0 = 2 2 Per = IL: –2

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**Ex 1) Graph b = 2 c = 0 a = 1 d Per = IL: Now graph on same axes**

Check Period: b c a d = 2 = 0 = 1 2 1 Per = IL: –1 –2 Continue graph to match cosine’s period

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**Ex 1) Graph Now add the 2 graphs together (y-values)**

At x = 0 = 2 At x = = 2.4 At x = = 0 At x = –1 + –1.4 = –2.4 At x = π 0 + –2 = –2 At x = 1 + –1.4 = –.4 3 2 1 –1 –2 –3

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**Let’s use Desmos to quickly graph the sum function and explore the period.**

Open Desmos. Choose , then Trigonometry , and then All the Trig Functions , then Explore this Graph! Tap into box 7 and delete boxes 4 – 7 add f (x) = to box 2 (use ) g (x) = to box 3

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Ex 2) Find the period of Check it with Desmos. sin x: put in front of x cos x: put in front of x 12π 18π 12π 8π 12π turn both graphs on

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In box 4, type f (x) + g (x) So confusing! The period should be 12π Turn off & 3 Stretch y-axis to [–4, 4] Pinch x-axis until you can see the graph repeat itself At x = 0, y = 1 Does it = 1 x = 12π? YEP!! turn it on

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**Ex 3) Graph y = sin x – x using ordinate addition **

We don’t always just graph trig + trig, sometimes we graph trig + polynomial Ex 3) Graph y = sin x – x using ordinate addition (by hand from –2π to 2π) Confirm/ Check with Desmos f (x) = sin x g (x) = –x Try On Your Own keep f (x) + g (x) f (x) = sin x g (x) = –x f (x) + g (x) sin x – x

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Homework # Pg 211 #9, 11, 12, 21, 22, 25, 29, 31, 34, 37, 38, 39, 40, 43, 49

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