4.1 – Graphs of the Sine and Cosine Functions

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Presentation transcript:

4.1 – Graphs of the Sine and Cosine Functions Math 150 4.1 – Graphs of the Sine and Cosine Functions

A periodic function is a function 𝑓 such that 𝑓 𝑥 =𝑓(𝑥+𝑛𝑝) for every real # 𝑥 in the domain of 𝑓, every integer 𝑛, and some positive real number 𝑝. The least possible positive value of 𝑝 is the period of the function.

Note: sin 𝑥 and cos 𝑥 are periodic with period _____.

Note: sin 𝑥 and cos 𝑥 are periodic with period _____. 𝟐𝝅

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Let’s try graphing 𝑦= sin 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Now let’s graph 𝑦= cos 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Ex 1. Graph 𝑦=2 sin 𝑥 .

Note: The amplitude of a periodic function is half the difference between the maximum and minimum values. For 𝑦=𝑎 sin 𝑥 and 𝑦=𝑎 cos 𝑥 the amplitude is 𝑎 .

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

Note: The period of both 𝑦= sin 𝑏𝑥 and 𝑦= cos 𝑏𝑥 is 2𝜋 𝑏 .

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.