Basic Ideas 1. Using a sketch to help, explain the connection between a circle with a point rotating on its circumference and a sinusoidal graph.

Transcription

1 Name Date Class # Block Intro to Calculus 1 Goals Modeling situations using trigonometric equations. Basic Ideas 1. 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 equations. 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. 1 These materials are based on CPM Pre-Calculus p Critical and Informed Thinkers Effective Communicators Collaborative Workers page 1

2 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 ) Critical and Informed Thinkers Effective Communicators Collaborative Workers page 2

3 Problems 1. Match each equation with one of the graphs. Every graph has a corresponding equation. a. y = cos ( 3x ) b. y = 2 sin ( x ) 1 c. y = cos ( 2x ) d. y = sin x 2 π e. y = sin x 4 f. y = cos ( x ) 2 g. y = 2 sin ( 2x ) h. y = 2 cos ( x + π ) Critical and Informed Thinkers Effective Communicators Collaborative Workers page 3

4 2. Predict the period for each of the equations below. Graph to check. a. y = sin 4x ( ) b. y = sin x 3 c. y = sin π 2 x d. y = sin 2π 7 x 3. What transformations are needed to change the graph of y = sin x ( ) into the graph shown? Describe 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 require a horizontal shift. a. b. Critical and Informed Thinkers Effective Communicators Collaborative Workers page 4

5 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. Critical and Informed Thinkers Effective Communicators Collaborative Workers page 5

6 7. Joey rode his bike over a piece of gum. Joey continued riding 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 equation 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. Critical and Informed Thinkers Effective Communicators Collaborative Workers page 6