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Exercises: Choosing Trigonometric Functions to Model Periodic Phenomena

Show your work for each problem. Express answers involving pi using exact form (e.g., pi/4) unless otherwise stated.

Grade 9·24 problems·~30 min·Common Core Math - HS Functions·standard·hsf-tf-b-5
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A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

In the function y=sin⁡(x)y = \sin(x), what is the period?

2.

Which transformation moves the graph of y=sin⁡(x)y = \sin(x) up 3 units?

3.

At t=0t = 0, a quantity is at its maximum value and then decreases. Which function best describes this starting behavior?

B

Fluency Practice

Practice the core skills of this lesson.

1.

What is the amplitude of y=4sin⁡(3x)−2y = 4\sin(3x) - 2?

2.

What is the period of y=cos⁡ ⁣(π6x)+1y = \cos\!\left(\frac{\pi}{6}x\right) + 1?

Express your answer as a whole number.

3.

What is the midline of y=−5cos⁡(2x)+7y = -5\cos(2x) + 7?

Give the yy-value of the midline as a whole number.

4.

A periodic phenomenon has a maximum value of 40 and a minimum value of 10. What is the amplitude?

Parameter extraction table for a tidal height model: max 6 ft, min 0 ft, period 12.5 hr, midline 3 ft, amplitude 3 ft, B unknown
5.

A tide has a maximum height of 6 feet and a minimum height of 0 feet, completing one full cycle every 12.5 hours.

What is the value of BB in the sinusoidal model? Enter BB as a decimal rounded to three decimal places.

6.

A pendulum displacement starts at its maximum positive value at t=0t = 0. The amplitude is 8 cm, the period is π\pi seconds, and the midline is y=0y = 0. Which equation models this?

C

Varied Practice

Practice the same skills in different formats and representations.

1.

A sinusoidal function has a maximum of 9 and a minimum of 1, and completes one cycle over an interval of length 4π4\pi. Which function matches these features?

Ferris wheel diagram showing maximum height 52 ft at the top and minimum height 2 ft at the bottom, with a dashed midline to be determined
2.

A Ferris wheel has a maximum height of 52 feet and a minimum height of 2 feet.

What is the midline (vertical center) of the height function, in feet? Give your answer as a whole number.

3.

A buoy bobs up and down. At t=0t = 0 it is at its midline moving upward. The amplitude is 3 ft, period is 8 s, and midline is y=5y = 5.

The model has the form y=Asin⁡(Bt)+Dy = A\sin(Bt) + D. Fill in the values: $A = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   , $D = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   , and $B = $   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲  

amplitude A:
midline D:
frequency factor B:
4.

A solar panel angle starts at its maximum of 90° at t=0t = 0 and decreases to a minimum of 0°, completing one full cycle in 24 hours. Which function is the simplest model?

5.

In the model h(t)=−25cos⁡ ⁣(π4t)+27h(t) = -25\cos\!\left(\frac{\pi}{4}t\right) + 27, what does the value 27 represent in the context of a Ferris wheel where hh is height in feet and tt is time in minutes?

6.

Using the Ferris wheel model h(t)=−25cos⁡ ⁣(π4t)+27h(t) = -25\cos\!\left(\frac{\pi}{4}t\right) + 27, what is the height (in feet) at t=4t = 4 minutes?

Give your answer as a whole number.

D

Word Problems

Apply your skills to real-world situations. Show your work.

1.

The average monthly temperature in a city ranges from a high of 85°F in July to a low of 25°F in January. Assume the temperature follows a sinusoidal model with a 12-month period.

What is the amplitude of the temperature model, in degrees Fahrenheit? Give your answer as a whole number.

Ferris wheel with center 30 ft above ground, radius 25 ft, showing boarding position at bottom (5 ft) and top (55 ft)
2.

A Ferris wheel has a diameter of 50 feet. Its center is 30 feet above the ground. Riders board at the bottom of the wheel. The wheel completes one full revolution every 6 minutes.

1.

What is the midline of the height function? Give your answer in feet as a whole number.

2.

Since riders board at the bottom at t=0t = 0, which function correctly models the height?

3.

Using the model h(t)=−25cos⁡ ⁣(π3t)+30h(t) = -25\cos\!\left(\frac{\pi}{3}t\right) + 30, what is the height of a rider after 1.5 minutes?

Give your answer in feet as a whole number.

3.

Ocean water temperature at a buoy follows a sinusoidal model. The temperature reaches a maximum of 22°C and a minimum of 14°C, completing one cycle every 24 hours. At t=0t = 0 hours, the temperature is at its maximum.

Write a cosine model T(t)=Acos⁡(Bt)+DT(t) = A\cos(Bt) + D for the temperature, then evaluate it at t=6t = 6 hours.

What is the temperature in °C at t=6t = 6 hours? Give your answer as a whole number.

E

Error Analysis

A student made an error in each problem below. Identify the mistake.

1.

Mia is given y=5sin⁡(3x)+2y = 5\sin(3x) + 2 and writes:

"The period is B=3B = 3, the amplitude is 5, and the midline is y=2y = 2."

What error did Mia make?

2.

Marcus models a Ferris wheel with maximum height 48 ft and minimum height 8 ft. He writes:

"Amplitude =−20= -20 because the function starts at the minimum and I used −cos⁡-\cos, so A=−20A = -20."

What is wrong with Marcus's reasoning?

F

Challenge / Extension

These problems go beyond the core lesson. Try them if you want an extra challenge.

1.

A sound wave has pressure function P(t)=0.4cos⁡(880πt)P(t) = 0.4\cos(880\pi t) where PP is in pascals and tt is in seconds.

A second sound wave has twice the amplitude and half the frequency of the first. What is the period of the second wave, in seconds? Give your answer as a fraction in simplest form.

2.

Two students model the same Ferris wheel (max height 52 ft, min height 2 ft, period 8 min, riders board at the midline going upward):

  • Student A: h(t)=25sin⁡ ⁣(π4t)+27h(t) = 25\sin\!\left(\frac{\pi}{4}t\right) + 27
  • Student B: h(t)=−25cos⁡ ⁣(π4t)+27h(t) = -25\cos\!\left(\frac{\pi}{4}t\right) + 27

Are both models correct for the given starting condition? Explain by checking each at t=0t = 0.

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