AP Precalculus · Free-response skill

Modeling a periodic context

The move. From facts about an invented periodic situation, find the amplitude, midline, period and phase, write a sinusoidal model, and use it to find a value and a time.

How to do it

  1. List the facts first: the highest and lowest values and the times they happen. A high and the next low are half a period apart.
  2. Amplitude is half the gap between high and low, and the midline is their average. Give each with the unit of the output, and give the period with the unit of time.
  3. Get b from the period with b=2πperiodb = \frac{2\pi}{\text{period}}. If you know when the maximum happens, use cosine shifted to that time: acos⁡(b(t−tmax⁡))+da\cos(b(t - t_{\max})) + d. If you know a midline crossing on the way up, use sine shifted to that time.
  4. Check the formula: put in the time of the maximum and make sure you get the maximum.
  5. To find a value, substitute and use special angles. To find a time, set the formula equal to the value, isolate the cosine or sine, use the unit circle, and pick the solution that fits the time window the question names.
  6. This question type sits in the no-calculator part of the free-response section, so practice every step with exact values.

Practice prompt Try it

At a fictional harbor called Gull Point, the depth of the water rises and falls with the tide in a regular cycle. On one day, high tide brings the depth to 14 feet at 2 a.m., and the next low tide brings it to 6 feet at 8 a.m. The depth D(t)D(t), in feet, is modeled by a sinusoidal function of t, the number of hours after midnight.

Use the facts about the tide to answer each part. Do not use a calculator.

  1. Work out the amplitude, the midline and the period of D. Give each one with its unit.
  2. Write a formula for D(t)D(t) as a sine or cosine function of t.
  3. Use your formula to work out the exact depth of the water at 6 a.m. Show the value you put into the formula.
  4. A fishing boat needs a depth of at least 12 feet to leave the dock. After the 2 a.m. high tide, at what time does the depth first fall to 12 feet? Give an exact answer, both as a number of hours after midnight and as a clock time.

Write each part, then check yourself against the checklist. Nothing you type here is saved or graded.

Check yourself and see a model answer

Checklist

Our own checklist of what a full answer does, one line per point. Tick each one your response earns.

0 of 7 points

Model answer, annotated

(a) The amplitude is 14−62=4\frac{14 - 6}{2} = 4 feet. The midline is D=14+62=10D = \frac{14 + 6}{2} = 10 feet. High tide at 2 a.m. and the next low tide at 8 a.m. are half a period apart, so the period is 2(8−2)=122(8 - 2) = 12 hours.

  • Point 1 Computes the amplitude and midline from the high and low depths and labels both in feet.
  • Point 2 Doubles the 6 hours from high to low tide and labels the period in hours.

(b) From the period, b=2π12=π6b = \frac{2\pi}{12} = \frac{\pi}{6}. The maximum is at t=2t = 2, and cosine starts at its maximum, so D(t)=4cos⁡(π6(t−2))+10D(t) = 4\cos\left(\frac{\pi}{6}(t - 2)\right) + 10. Check: D(2)=4cos⁡0+10=14D(2) = 4\cos 0 + 10 = 14.

  • Point 3 Turns the period of 12 hours into the coefficient π/6.
  • Point 4 Shifts cosine to the time of high tide and checks that D(2) is 14.

(c) At 6 a.m., t=6t = 6: D(6)=4cos⁡(π6⋅4)+10=4cos⁡(2π3)+10=4(−12)+10=8D(6) = 4\cos\left(\frac{\pi}{6} \cdot 4\right) + 10 = 4\cos\left(\frac{2\pi}{3}\right) + 10 = 4\left(-\frac{1}{2}\right) + 10 = 8 feet.

  • Point 5 Shows the substitution, uses the exact value of cos(2π/3) and labels the answer in feet.

(d) Set 4cos⁡(π6(t−2))+10=124\cos\left(\frac{\pi}{6}(t - 2)\right) + 10 = 12, so cos⁡(π6(t−2))=12\cos\left(\frac{\pi}{6}(t - 2)\right) = \frac{1}{2}. Right after high tide the angle starts at 0 and grows, and the first angle with cosine 12\frac{1}{2} is π3\frac{\pi}{3}. So π6(t−2)=π3\frac{\pi}{6}(t - 2) = \frac{\pi}{3}, which gives t−2=2t - 2 = 2 and t=4t = 4. The depth first falls to 12 feet 4 hours after midnight, at 4 a.m.

  • Point 6 Writes the equation and isolates the cosine before solving.
  • Point 7 Picks the first angle after high tide and gives the answer in both forms the part asks for.