13-5The cosine Function

Today’s Objective:

I can graph the cosine function.

𝑦=co s 𝑥

𝑓 (𝑥 )=sin 𝑥x y

0

x

y

-3p -5p

2-2p -3p

2-p -p

2

p

2p 3p

22p 5p

23p

-2

-1

1

2

x

y

Domain:

Range:

Period:

Amplitude:

All real numbers

– 1 ≤ y ≤ 1

2π

1

𝑦=cos 𝑥 𝑦=𝑐𝑜𝑠 𝜃OR

-3p -5p

2-2p -3p

2-p -p

2

p

2p 3p

22p 5p

23p

-2

-1

1

2

x

y

How does this graph differ from y = sin x?

Translated left by

𝜋2

x

y

Properties of Cosine Function

𝑦=𝒂cos𝒃𝑥 b = # of cycles from 0 to 2πPeriod of function =

= amplitude– a = reflects graph across x-axis

Sketch the graph of:

𝑦=𝟏 .𝟓cos𝟐 𝑥 𝑦=𝟐cos𝑥𝟑

p

2p 3p

22p

-2

-1

1

2

x

y

p

2p 3p

22p

-2

-1

1

2

x

y

p

2p 3p

22p

-2

-1

1

2

x

y

p

2p 3p

22p

-2

-1

1

2

x

y

x

y

p 2p 3p 4p 5p 6p-2

-1

1

2

x

y

p 2p 3p 4p 5p 6p-2

-1

1

2

x

y

p 2p 3p 4p 5p 6p-2

-1

1

2

x

y

5 points: max – midline – min – midline - max

What is the cosine function that models the water level in inches as a function of time?

Amplitude:12

(60 )=¿30Period:

4:40 pm – 10:30 am hrs

1213=

2𝜋𝑏

𝑦=𝒂cos𝒃𝑥

𝑏=6𝜋37

𝑓 (𝑡 )=−𝟑𝟎cos𝟔𝝅𝟑𝟕

𝑥

𝑎=−30

2()=¿

My boat needs 10 inches above an average water level. When can I bring my boat into this dock?

p. 864: 7-19 odd, 27-33