Ex #1 - MS CLINE'S MATH CLASSES



Grade 12

Pre-Calculus Mathematics

Notebook

Chapter 5

Trigonometric

Functions and

Graphs

Outcomes: T4

[pic]

5.1 Graphing Sine and Cosine Functions T4

(p.222-232)

Periodic Functions = A function that repeats itself over regular intervals

(cycles) of its domain.

Period = The length of the interval over which a function travels one full

rotation or cycle. (The distance from one point to that exact same

point again.)

Amplitude = the vertical distance from the horizontal central axis

the maximum or the minimum. Note that the central axis is not

always the x-axis.

Sinusoidal Functions = Oscillating periodic functions, like[pic] or

[pic].

[pic]

|x |  |

|Central axis | |

|Amplitude | |

|Period | |

|Phase shift | |

|Maximum | |

|Minimum | |

|Range | |

Try these:

a) [pic]

b) [pic]

Homework: Page 250 #1, 2, 3

5.2 Transformations of Sinusoidal Functions – Part 2 T4

(p.238-249)

A sinusoidal function is expressed in the form:

[pic] or [pic]

|Properties of the graph |Calculations |

| a | |

| d | |

| Period | |

| b | |

| c = sine funcion | |

| c = cosine function | |

Write the equation of the following graphs in the form

[pic] or [pic] 

a)

b)

c) The point (–2,7) is a maximum of a sinusoidal function.

The next consecutive maximum occurs at the point (6, 7).

The range of the function is [–1,7].

d) This one is tough. Give it a try if you would like a challenge.

5.2 Writing Equations: T4

Transformations of Sinusoidal Functions

Write an equation for each of the following in terms of y = asin(b(x - c)) + d

Write an equation for each of the following in terms of y = acos(b(x – c)) + d

Homework: Page 250 #4-10, 13, 14-16

[pic]

[pic]

5.4 Equations and Graphs of Trigonometric Functions T4

(p.266-274)

Ex1: a) Sketch the following graph over the interval [pic]:

[pic]

b) Solve the following equation algebraically:

[pic]

c) Solve the same equation graphically.

d) Use the graph to explain why the following equation has no solution:

[pic]

Ex2: The graph of [pic] is sketched below.

[pic]

Graphically solve the equation [pic] over the interval [pic].

Note:

To find the zeros of [pic], we produce the equation [pic].

This equation simplifies to [pic]. Thus, the zeros are the solutions.

Ex3: The average daily maximum temperature in Winnipeg follows a sinusoidal pattern. The lowest value of –14°C is found on January 15 and the highest temperature is 26°C on July 15.

Key to this question: January 15 → 15th day of the year

July 15 → 196th day

There are 365 days in one year

a) Write a sinusoidal equation to represent this scenario using the cosine function.

b) Find the average temperature on October 27 (300th day).

Homework: Page 275 #5, 9, 15, 17, 18, 19, 20, 21

Applications of Periodic Functions T4

Problem 1

At a seaport the depth of the water, h meters, at time, t hours, during a certain day is given by this formula.

[pic]

a) State the:

i) period ii) amplitude iii) phase shift

b) What is the maximum depth of the water? When does it occur?

c) What is the depth of the water at 5:00 am?

Problem 2

A tsunami is a very fast-moving ocean wave caused by earthquakes that occur underwater. The water will first move down from its normal level, then move an equal distance above the normal level, then finally back to the normal level. The period of a tsunami is 16 minutes with amplitude of 8 meters.

The normal depth of water at Crescent Beach is 6 meters.

a) What is the maximum and minimum height of water caused by the tsunami at Crescent Beach?

b) Write a sinusoidal function to represent the tsunami when it first reaches Crescent Beach.

Assignment

Answer the following questions on a separate piece of paper.

1) The pedals on a bike have a maximum height of 30cm above the ground and a minimum distance of 8cm above the ground. A person pedals at a constant rate of 20 cycles per minute.

a) What is the period in seconds for this periodic function?

b) Determine an equation for this periodic function.

2) A ferris wheel of radius 25 meters, placed one meter above the ground varies sinusoidally with time. The ferris wheel makes one rotation every 24 seconds, with a person sitting 26 meters from the ground and rising when it starts to rotate.

a) Write a sinusoidal function that describes the function from a person’s starting point.

b) How high above the ground would a person be 16 seconds after the ferris wheel starts moving?

3) Tides are a periodic rise and fall of water in the ocean. A low tide of 4.2 meters in White Rock, B.C. occurs at 4:30 am, and the next high tide of 11.8 meters occurs at 11:30 am the same day.

a) Write a sinusoidal function that describes the tide flow.

b) What is the tide height at 1:15 pm that same day?

4) A spring modeling a sinusoidal function rests 1.6 meters above the ground. If the mass on the spring is pulled 1.1 meters below its resting positions and then released, it requires 0.5 seconds to move from the minimum positions to its maximum positions. Assuming friction and air resistance are neglected, write an equation in terms of cosine that describes this periodic function.

5) A sinusoidal function has a maximum at [pic] and the next nearest minimum is at [pic]. Write an equation to represent this graph.

-----------------------

x

y

y

x

x

y

x

y

x

y

x

y

x

y

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

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