AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/3/3-7)
Unit 3 · Topic 3.7
3.7 Sinusoidal Function Context and Data Modeling
Repeating real-world data, such as daily temperatures over a year, the height of a Ferris wheel car or the depth of water at a dock, can be modeled with a sinusoidal function. You build the model from the max, min, period and one known point, or with sinusoidal regression, then use it to predict.
Key terms
- sinusoidal model
- sinusoidal regression
- context
- prediction
- periodic data
Building a model by hand
Read four facts from the context or data:
- Period: the time between consecutive maximums (or minimums). Then b = 2π/period.
- Midline: d = (max + min)/2.
- Amplitude: |a| = (max − min)/2.
- Phase shift: compare a known input-output pair with the model. If you know when a maximum happens, a cosine model shifted to that time is easiest. If the quantity starts at a minimum, use a negative cosine.
Choosing sine or cosine
Both work, so choose the one that makes the shift simplest.
- Starts at a maximum: a cos(bt) + d with a > 0.
- Starts at a minimum: −|a| cos(bt) + d.
- Starts on the midline and rising: a sin(bt) + d with a > 0.
- Starts on the midline and falling: −|a| sin(bt) + d.
Sinusoidal regression
With data, a graphing calculator can fit a sinusoidal regression (often called SinReg). Put the calculator in radian mode first. The regression returns a model of the form a sin(bx + c) + d.
A good estimate of the period can help the regression find a sensible fit. Check the result: its period and midline should match what you see in the data.
Using the model
Predict outputs by plugging in inputs. Find when the output reaches a value by solving the equation; on a calculator, graph the model and the horizontal line y = target and find the intersection points.
Remember that a periodic equation has many solutions, so list every one in the time window you care about.
Models are usually only valid over a contextual domain. A model of tide heights for one week shouldn't be trusted for next year, because real tides drift.
Checking and interpreting a model
Before you use a model, test it. Plug in the time of a known maximum or minimum and make sure you get the right value. Check that 2π/b equals the period you read from the context.
Then say what each constant means. In T(t) = 29 cos((π/6)(t − 6.5)) + 59 from the example below, 59 is the average high temperature across the year, 29 is how far the temperature swings above or below that average, the period of 12 months is one year, and 6.5 is when the hottest point of the year occurs.
Worked examples
Try each one yourself first, then open the solution.
- Example 1
A Ferris wheel
A Ferris wheel has a diameter of 40 meters, and its center is 25 meters above the ground. It turns once every 8 minutes. A rider boards at the lowest point at t = 0. Write a model for the rider's height h(t) in meters after t minutes, and find the exact height at t = 3.
Show the solutionHide the solution
- Step 1: Midline: the center height, d = 25. Amplitude: the radius, 20.
- Step 2: Period 8 minutes, so b = 2π/8 = π/4.
- Step 3: The rider starts at the minimum, so use a negative cosine: h(t) = −20 cos(πt/4) + 25.
- Step 4: h(3) = −20 cos(3π/4) + 25 = −20(−√2/2) + 25 = 25 + 10√2.
- Step 5: That's about 39.142 meters.
Answer: h(t) = −20 cos(πt/4) + 25; h(3) = 25 + 10√2 ≈ 39.142 meters.
- Example 2Calculator allowed
Temperature over a year
A city's average daily high temperature peaks at 88°F at t = 6.5 (mid-July), where t is months after January 1, and its lowest value is 30°F. The pattern repeats every 12 months. Write a model and find the part of the year when the average high is above 75°F.
Show the solutionHide the solution
- Step 1: Midline: (88 + 30)/2 = 59. Amplitude: (88 − 30)/2 = 29. b = 2π/12 = π/6.
- Step 2: The maximum is at t = 6.5, so use cosine shifted right 6.5: T(t) = 29 cos((π/6)(t − 6.5)) + 59.
- Step 3: Solve T(t) = 75 by graphing y = T(t) and y = 75 in radian mode. The intersections in 0 ≤ t ≤ 12 are t ≈ 4.616 and t ≈ 8.384.
- Step 4: Between those times the graph is above 75, because the peak at t = 6.5 lies between them.
Answer: T(t) = 29 cos((π/6)(t − 6.5)) + 59; above 75°F for about 4.616 < t < 8.384, roughly late May to mid-September.
- Example 3
Trap: using the period as b
Water at a dock is 12 feet deep at high tide at midnight (t = 0) and 4 feet deep at the next low tide 6 hours later. A student writes D(t) = 4 cos(12t) + 8. Fix the model and find the depth at t = 2.
Show the solutionHide the solution
- Step 1: Midline (12 + 4)/2 = 8 and amplitude (12 − 4)/2 = 4 are correct, and cosine fits because the tide starts at a maximum.
- Step 2: High to low is half a period, so the period is 12 hours. But b is not the period: b = 2π/12 = π/6.
- Step 3: Correct model: D(t) = 4 cos(πt/6) + 8.
- Step 4: D(2) = 4 cos(π/3) + 8 = 4(1/2) + 8 = 10. The student's model would give about 9.697 feet, which is wrong.
Answer: D(t) = 4 cos(πt/6) + 8; the depth at t = 2 is 10 feet.
Common mistakes
- Writing the period where b belongs. Use b = 2π/period.
- Starting the model at the wrong point in the cycle. Check your equation at t = 0 against the context.
- Running a sinusoidal regression in degree mode.
- Giving only one solution when the context window holds several cycles.
On the exam
- The periodic-modeling free-response question (no calculator) typically gives a context and key points, asks you to find a, b, c and d, and asks what the model says at certain times. Show how you got each constant.
- That question may also ask for the coordinates of labeled points on the graph, such as maximums, minimums and midline crossings, over two full cycles. Neighboring key points are a quarter period apart.
- Interpret answers in context with units, such as “the depth is 10 feet at 2 a.m.”
Connected topics
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Check yourself
5 questions on 3.7 Sinusoidal Function Context and Data Modeling. Pick an answer to see if you got it, and why.
In a certain city, the average daily high temperature varies sinusoidally over the year. Let t be the time in months, where t = 1, 2, …, 12 stand for the middle of January, February, …, December. The average daily high reaches a maximum of 88°F in mid-July (t = 7) and a minimum of 40°F in mid-January (t = 1). The model T(t) = 24 cos(π(t − 7)/6) + 64 fits these facts. According to the model, which of the following is closest to the average daily high temperature at the end of April (t = 4.5)?
In a certain town, the number of hours of daylight varies sinusoidally over the year. The longest day has 15.2 hours of daylight and comes on day t = 172 of the year. The shortest day has 9.1 hours of daylight. Assume the pattern repeats every 365 days.
Invented scenario
Which of the following could be a model for D(t), the number of hours of daylight on day t?
Using the model D(t) = 3.05 cos((2π/365)(t − 172)) + 12.15, which of the following is closest to the number of hours of daylight on day t = 80?
A Ferris wheel has a diameter of 40 meters, and its center is 25 meters above the ground. The wheel turns at a constant speed and makes one full revolution every 8 minutes. A rider boards at the lowest point of the wheel at time t = 0 minutes. Let h(t) be the rider's height above the ground, in meters, at time t.
Invented scenario
Which of the following could define h(t)?
During the first revolution, for how many minutes is the rider more than 35 meters above the ground?
0 of 5 answered