Algebra 2: Modeling with Periodic Functions

The y-value for the midline is equal to ___________.

(max y+ min y)/2

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Which periodic function, sine or cosine, would be a simpler model for the situation? Explain. Which did you include in your response

-A cosine function would be a simpler model for the situation. -The minimum depth (low tide) occurs at t = 0. A reflection of the cosine curve also has a minimum at t = 0. -A sine model would require a phase shift, while a cosine model does not.

What do the graphs of sine and cosine have in common with the swinging you see?

-The high and low points repeat in a pattern. -The cycle repeats at equal time intervals. -The swinging motion is smooth, unabrupt

Which situations can be modeled with a periodic function?

-the height of ball suspended from a spring -the height of a flag on one blade of a windmill

The graph models the height of the end of a blade of a windmill as a function of time. Assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate. Use the graph to complete the statements. 1.The blades of the windmill turn on an axis that is ___ feet from the ground. 2.The blades of the windmill are ___ feet long. 3. The blade makes one revolution in ___ seconds.

1. 20 ft 2. 5 ft 3. 30 sec

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Write a cosine model, d = acos(bt) + k, for the depth as a function of time. 1. The amplitude is ______ meters. 2. a= 3. The period is_______ 4. b= 5. The vertical shift is k=____ meters. 6. Choose the correct equation.

1. amplitude 1.5 m 2. a= -1.5 3. The period is 12.5 hours 4. b= 4pi/25 5. The vertical shift is k= 4 meters 6. d= -1.5cos(4π/25t) +4

Use the graph to write an equation y= acos(b(x-h))+k to model the situation. 1.The amplitude= 2.a= 3. The period is 4. b= 5. Choose the equation of the graph

1. amplitude= 30 2. a=-30 3. period = 12 4. pi/6 5. y=-30cos(π/6(x-+1))+50

The blades of a windmill turn on an axis that is 40 feet from the ground. The blades are 15 feet long and complete 3 rotations every minute. Write a sine model, y = asin(bt) + k, for the height (in feet) of the end of one blade as a function of time t (in seconds). Assume the blade is pointing to the right when t = 0 and that the windmill turns counterclockwise at a constant rate. 1. a is the... 2. The vertical shift, k, is the... 3. the period is ____ seconds. 4. b is 5. Choose the equation of the model.

1. length of the blade (a=15) 2. height of the windmill (k=40) 3.20 4. pi/10 5. y=15sin(π/10t)+40

The equation d=11cos(8π/5t) models the horizontal distance, d, in inches of the pendulum of a grandfather clock from the center as it swings from right to left and left to right as a function of time, t, in seconds. According to the model, how long does it take for the pendulum to swing from its rightmost position to its leftmost position and back again? Assume that right of center is a positive distance and left of center is a negative distance.

1.25 seconds

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Graph the equation d= -1.5cos(4π/25t)+4 that models the situation using a graphing calculator and use it to answer the following questions. How many times during this day is the depth at the end of the pier equal to 4 meters?

4 times

The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let t = 0 be 12:00 am. Graph the equation d= -1.5cos(4π/25t)+4 that models the situation using a graphing calculator and use it to answer the following questions. At approximately what time on the next day does the depth first reach 4 meters?

4:00am

The equation h= can be used to model the height, h, in feet of the end of one blade of a windmill turning on an axis above the ground as a function of time, t, in seconds. How long is the blade? Assume that the blade is pointing to the right, parallel to the ground, at t = 0, and that the windmill turns counterclockwise at a constant rate.

7 ft

The height, h, in feet of the tip of the minute hand of a wall clock as a function of time, t, in minutes can be modeled by the equation h=0.75cos(π/30(t-15))+8. Which number (from 1 to 12) is the minute hand pointing to at t = 0?

9

The height, h, in feet of a flag on one blade of a windmill as a function of time, t, in seconds can be modeled by the equation h=3sin(4π/5(t-1/2))+12. What is the minimum height of the flag?

9 ft

Blood pressure can be modeled by a sinusoidal curve, where the maximum and minimum on the curve correlate to the person's blood pressure reading. Henry's blood pressure is modeled by the function P(t)=30sin(2πt)+100, where t is time in seconds. Which graph accurately depicts this model?

a

Starting at its rightmost position, it takes 1 second for the pendulum of a grandfather clock to swing a horizontal distance of 12 inches from right to left, and 1 second for the pendulum to swing back from left to right. Write a cosine function, d = acos(bt), to model the distance, d, of the pendulum from the center (in inches) as a function of time t (in seconds).

a= 6 The period is 2 seconds The equation of the model is: d = 6cos(πt) -Graph the function using the graphing calculator. Find the least positive value of t at which the pendulum is in the center. t = 0.5sec -To the nearest thousandth, find the position of the pendulum when t= 4.25 sec. d = 4.243in.

Derive an equation d = acos(bt) for the displacement d (in feet) in relation to sea level of a piece of cloth tied to a water wheel over t seconds. Assume the water wheel is half-submerged and takes 24 seconds to complete one turn. At t = 0 seconds, the cloth is at a height of 10 feet above sea level, its maximum displacement from the surface of the water. 1. When does the cloth first hit the water? 2.What is d when t = 4 sec?

a=10 b=π/12 equation: d=10cos(π/12t) 1. t=6sec 2.t=5ft

Derive an equation d = asin(bt) for the displacement d (in feet) of a buoy in relation to sea level over t seconds. (Assume the buoy starts with a displacement of 0 and then goes up to its maximum height.) The maximum displacement of the buoy is 1.5 feet in either direction, and the time it takes for the buoy to go from its highest point to its lowest point is 3 seconds.

amplitude: 1.5 b= pi/3 equation: d=1.5sin(π/3t)

Which graph could model the path of a pendant attached to the outer edge of a wheel rim with a 7-inch radius if it takes 4 seconds for the wheel to make 1 revolution, and the tire is 4 inches thick?

b

The table shows the average daily temperature on the first day of each month for one year. Use the graphing calculator to graph the data points. Use 0 for January, 1 for February, 2 for March, and so on. The graph of these points follows a path resembling a _______.

cosine curve reflected across the x-axis

A weight attached to a spring is at its lowest point, 9 inches below equilibrium, at time t = 0 seconds. When the weight it released, it oscillates and returns to its original position at t = 3 seconds. Which of the following equations models the distance, d, of the weight from its equilibrium after t seconds?

d=-9cos(2π/3t)

Starting at its rightmost position, it takes 2 seconds for the pendulum of a grandfather clock to swing a horizontal distance of 18 inches from right to left and 2 seconds for the pendulum to swing back from left to right. Which of the following equations models d, the horizontal distance in inches of the pendulum from the center as a function of time, t, in seconds? Assume that right of center is a positive distance and left of center is a negative distance.

d=9cos(π/2t)

The height, h, in feet of a ball suspended from a spring as a function of time, t, in seconds can be modeled by the equation h=-sin(π(t+1/2))+5. Which of the following equations can also model this situation?

h=-2cos(πt)+5

A buoy starts at a height of 0 in relation to sea level and then goes up. Its maximum displacement in either direction is 6 feet, and the time it takes to go from its highest point to its lowest point is 4 seconds. Which of the following equations can be used to model h, the height in feet of the buoy in relation to sea level as a function of time, t, in seconds?

h=6sine(pi/4t)

-cosine

min-mid-max-mid-min

A driver runs over a nail, puncturing the tire without causing a leak. The position of the nail in the tire, with relation to the ground, while the car is moving at a constant speed is shown in the table. Which key feature of the function representing the nail's travel can be used to determine the amount of time it takes for the nail to reach the same orientation it had when it entered the tire?

period