4.5(d) Notes: Modeling Periodic Behavior Date: 4.5(d) Notes: Modeling Periodic Behavior   Lesson Objective: To find sine and cosine functions. CCSS: F-TF Extend the domain of tri­go­no­me­tric functions using the unit circle. You will need: graphing calculator

Lesson 1: Using a Graph to Find an Equation   Find an equation of the form y = A sin Bx that produces the graph shown in the figure.

Lesson 2: Finding an Equation from Data A region that is 30° north of the Equator aver­ages a minimum of 10 hours of daylight in De­cember. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 representing January, etc. If y repre­sents the number of hours of daylight in month, x, use a sine function of the form y = A sin(Bx – C) + D to model the hours of daylight. Graph the function. 

Lesson 2: Finding an Equation from Data A region that is 30° north of the Equator aver­ages a minimum of 10 hours of daylight in De­cember. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 representing January, etc. If y repre­sents the number of hours of daylight in month, x, use a sine function of the form y = A sin(Bx – C) + D to model the hours of daylight. Graph the function. Min = 10 hrs in De­c., Max = 14 hrs in June January = 1, Period = 2(Dec. – June) = 2(6) = 12

Lesson 2: Finding an Equation from Data Min = 10 hrs in De­c., Max = 14 hrs in June January = 1, Period = 2(Dec. – June) = 2(6) = 12 y = number of hours of daylight in month, x  

Lesson 2: Finding an Equation from Data D = Max + Min = Interval = Period = 2 4 |A| = Max – Min = Period = 2π 2 B Phase Shift = C B  

Lesson 3: More Modeling   The depth of water at a boat dock varies with the tides. The depth is 5’ at low tide and 13’ at high tide. On a certain day, low tide is at 4 AM and high tide is at 10 AM. If y represents the depth of the water, in feet, x hours after mid­night, use a sine function of the form y = A sin(Bx – C) + D to model the water’s depth. Graph the function.

Lesson 3: More Modeling The depth of water at a boat dock varies with the tides. The depth is 5’ at low tide and 13’ at high tide. On a certain day, low tide is at 4 AM and high tide is at 10 AM. If y represents the depth of the water, in feet, x hours after mid­night, use a sine function of the form y = A sin(Bx – C) + D to model the water’s depth. Graph the function. The depth is 5’ at low tide and 13’ at high tide. Low tide is at 4 AM and high tide is at 10 AM. Period = 2(10 – 4) = 12

Lesson 3: More Modeling The depth is 5’ at low tide and 13’ at high tide. Low tide is at 4 AM and high tide is at 10 AM. Period = 2(10 – 4) = 12  

D = Max + Min = Interval = Period = 2 4 |A| = Max – Min = Period = 2π Lesson 3: More Modeling D = Max + Min = Interval = Period = 2 4 |A| = Max – Min = Period = 2π 2 B Phase Shift = C B  

Find the equation for the graph. 4.5(d): Do I Get It? Yes or No   Find the equation for the graph.

4.5(d): Do I Get It? Yes or No  

a. Use a trigonometric function to model the data. 4.5(d): Do I Get It? Yes or No   3. Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine varies with the tides. The table shows the depths (in feet) at various times during the morning. a. Use a trigonometric function to model the data. b. Find the depths at 9 A.M. and 3 P.M. c. A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon can it safely dock?

a. Use a trigonometric function to model the data. 4.5(d): Do I Get It? Yes or No   a. Use a trigonometric function to model the data.

b. Find the depths at 9 A.M. and 3 P.M. 4.5(d): Do I Get It? Yes or No   b. Find the depths at 9 A.M. and 3 P.M. c. A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon can it safely dock?