BASIC FUNCTION



Use unit circle to complete the following table and then graph y= sin x and y= cos x

x |-2ᴫ |-3 ᴫ/2 |-ᴫ |-ᴫ/2 |0 |ᴫ/2 | ᴫ |3ᴫ/2 |2ᴫ | |Sin x | | | | | | | | | | |Cos x | | | | | | | | | | |

BASIC FUNCTION : [pic]

Domain: __________________

Range: ___________________

Continuous and Periodic

Symmetry: ________________

No asymptotes

End behavior: The function oscillates between -1 and 1

What is the period of [pic]?

What is the amplitude of [pic]?

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BASIC FUNCTION : [pic]

Domain: __________________

Range: ___________________

Continuous and Periodic

Symmetry: ________________

No asymptotes

End behavior: The function oscillates between -1 and 1

What is the period of [pic]?

What is the amplitude of [pic]?

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A function is a sinusoid if it can be written in the form

[pic]

Where A, b, c and d are constants and neither A nor b is 0.

Is [pic] a sinusoid? __________________________________________________

______________________________________________________________________

The amplitude of a sinusoid is |A|. Graphically, the amplitude is half the height of the wave. If d =0, the amplitude is the distance from the x-axis to the max y-value.

What is the amplitude of each of the following?

[pic] ________ [pic] ________ [pic] ________

The period of the sinusoids [pic] is [pic]. Graphically, the period is the length of one full cycle of the wave or the length of one repetition of the pattern.

What is the period of each of the following?

[pic] ________ [pic] ________ [pic] ________

The frequency of the sinusoids [pic] is [pic]. Graphically, the frequency is number of complete cycles the wave completes in a unit interval.

The phase shift of the sinusoids [pic] is [pic]. Graphically, the phase shift is a horizontal translation of the wave. d is the vertical translation.

Example:

Construct a sinusoid that rises from a minimum value of [pic] at [pic] to a maximum value of [pic] at [pic].

Amplitude = ________________

Period = ____________________

k = _________ c = __________ d = ___________

Equation: __________________________________

Example:

One particular day in Galveston, TX, high tide occurred at 9:36 a.m. At that time the water at the end of the pier was 2.7 m deep. Low tide occurred at 3:48 p.m. at which time the water was only 2.1 m deep. Assume that the depth of the water is a sinusoidal function of time with a period of half a lunar day or about 12 hours 24 minutes.

A) At what time on that day did the first low tide occur/

B) What was the approximate depth of the water at 6:00 a.m. and at 3:00 p.m. that day?

C) What was the first time that day when the water was 2.4 m deep

Accommodations

Practice:

Find the amplitude of the function and use the language of transformations to describe how the graph of the function is related to either the graph of [pic] or [pic].

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

Find the amplitude, period and frequency of the function and use this information (not your calculator) to sketch a graph of the function in the window [-3(, 3(] by [-4, 4].

6. [pic]

`

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. A Ferris wheel, 50 ft in diameter, makes one revolution every 40 seconds. If the center of the wheel is 30 ft above the ground, how long after reaching the low point is a rider 50 ft above the ground?

12. An earthquake occurred at 9:40 a.m. on November 1, 1755 in Lisbon, Portugal, starting a tsunami (tidal wave) in the ocean. It produced waves that traveled more than 540 ft/s (370 mph) and reached a height of 60 ft. If the period of the waves was 30 min or 1800 seconds, estimate the length L between the crests of the waves.

13. One Labor Day in southern California the high tide occurred at 7:12 a.m. At that time the water at the pier was 11 ft deep. At 1:24 p.m. it was low tide and the water at the pier measured only 7 ft. Assume the depth of the water is a sinusoidal function of time with a period of ½ lunar day or 12 hours 24 minutes.

A) At what time on that Labor Day does the first low tide occur?

B) What was the approximate depth of the water at 4 a.m. and at 9 p.m.?

C) What is the first time on that Labor Day that the water is 9 ft. deep?

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