Intro to CalculusLesson for Homework 211

Basic Ideas1. Using a sketch to help, explain the connection between a circle with a point rotating

on its circumference and a sinusoidal graph.

2. a. Describe a few specific situations which might be modeled by trigonometric equa-tions. Make sure your descriptions include the variables.

b. What are the common characteristics of the situations described by you above and those described by the class in general.

Name ________________ Date _______ Class # ______ Block ______

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1 These materials are based on CPM Pre-Calculus p.449-453

GoalsModeling situations using trigonometric equations.

3. How could the graphs of y = sin x( ) (or y = cos x( ) ) be changed? Make a sketch of each possibility and describe what is happening.

4. Describe the transformations needed to change the given base graph into the graph shown. Write an equation and check using a graphing calculator.

f x( ) = cos x( ) f x( ) = sin x( )

Lesson for Homework 21

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Problems1. Match each equation with one of the graphs. Every graph has a corresponding

equation.

a. y = cos 3x( ) b. y = 2sin x( ) −1 c. y = cos 2x( ) +1

d. y = sin 12x⎛

⎝⎜⎞⎠⎟ e. y = sin x −

π4

⎛⎝⎜

⎞⎠⎟ f. y = cos x( ) − 2

g. y = −2sin 2x( ) h. y = 2cos x + π( )

Lesson for Homework 21

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2. Predict the period for each of the equations below. Graph to check.

a. y = sin 4x( ) b. y = sin x3

⎛⎝⎜

⎞⎠⎟

c. y = sin π2x⎛

⎝⎜⎞⎠⎟ d. y = sin 2π

7x⎛

⎝⎜⎞⎠⎟

3. What transformations are needed to change the graph of y = sin x( ) into the graph shown? De-scribe these transformations using words and an equation. Verify with a graphing calculator.

4. Repeat problem 3 using y = cos x( ) as the base function.

5. For each graph below, find an equation using sine and an equation using cosine that will generate the graph. For each graph, one of the functions you find should not re-quire a horizontal shift.

a. b.

Lesson for Homework 21

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6. Briele and Elaine always travel to Sunset Beach together for their summer vacation. Through the years they return to the same section of beach and spend their time around the pier. Elaine has noticed that the water rises and falls on the posts of the pier. She thinks the height of the water could be described as a function of time. Brielle was snorkeling and noticed that the pier posts are slimy up to the lowest level of the tide, which was 2 feet from this particular post. Since Brielle and Elaine were on vacation, they had nothing better to do an found out there was a low point on the post at eight o'clock in the morning and a high point 40 inches higher at two o'clock in the afternoon. Find an equation for the height of the water relative to the post as a function of time form 12 o'clock midnight.

a. At 11:30 a.m. the girls wanted to take a walk on the beach but decided to wait until high tide. What was the height of the water at 11:30 a.m.? Round to the nearest 0.1 inch.

b. Brielleʼs little brother Sean is 2 feet 7 inches tall. When is the first time after 8:00 a.m. that the water will be “over his head” if he plays right next to the post? Round to the nearest minute.

Lesson for Homework 21

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7. Joey rode his bike over a piece of gum. Joey continued rid-ing at a constant rate. At time t = 1.25 seconds, the gum was at a maximum height above the ground and 1 second later the gum was at a minimum. If the diameter of the wheel (all the way out to the edge of the tire) is 68 cm, create an equa-tion that will allow for the prediction of the height of the gum in centimeters at any time t.

a. Find the height of the gum when Joey gets to the corner at t = 15.6 seconds, assuming he maintains a constant speed. Round to the nearest 0.1 cm.

b. Find the first and second time the gum reaches a height of 12 cm while Joey is riding at a constant rate. Round to the nearest 0.1 second.

Lesson for Homework 21

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