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Algebra 2 9-12 / Week 10 / Monday
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Week 10 · Periodic Models

Monday

Three numbers, one wave
// The vent timer and the temperature wave
⏱ about 20 min

Monday: Three Numbers, One Wave

Sunset pours through the glass as Comet cranks the roof vents open, a grown-up steadying the ladder.

Wren spreads Nova's temperature log across the bench. "Every six hours, all week. What do you notice?"

Comet reads down the column. "20, 26, 20, 14, 20. Then it starts over."

"It goes up, comes down, dips under, and comes back," Wren says. "Like a wave."

Nova projects the five points joined by a smooth curve. "Would you like a hint? Week 9's sine curve did this too."

"Then we can write a rule for it," Comet says. "The high, the low, and how long before it repeats."

"Amplitude, midline, period," Wren says. "Three numbers, one wave. Our Glass House model, from our own log."

Sunset through the Glass House roof as Comet cranks the vents open and Wren plots a wave while Nova hovers.

The crew's vent-timer log

Every number in this table is the crew's own made-up reading from Nova's log. Hour 0 is the morning reading.

A pattern that repeats at a steady spacing is called periodic. The temperature wave repeats every day.

Hours after the morning readingTemperature (°C)
020
626
1220
1814
2420
The crew's temperature wave: midline 20 dashed, amplitude 6, period 24 hours, shown over two days.

A solved problem to study

Here is how Wren reads the three numbers from the table.

  1. The highest reading is 26 and the lowest is 14.
  2. Amplitude: half the distance from low to high. (26 - 14) ÷ 2 = 6.
  3. Midline: the average of high and low. (26 + 14) ÷ 2 = 20.
  4. Period: the wave is back at 20 and rising at hour 24. So the period is 24 hours.
  5. The wave starts at its midline and rises, like sin x. So the model is y = 6 sin(bx) + 20.
  6. The number b makes one full turn (2π) fit in 24 hours: b = 2π ÷ 24 = π/12.
  7. The crew's model: T(t) = 6 sin((π/12)t) + 20.

Check it at hour 6: (π/12) × 6 = π/2, and sin(π/2) = 1. So T(6) = 6 × 1 + 20 = 26. It matches the log.

A model is the crew's own fit to their readings. It is not a law about weather, and another week may need new numbers.

READ THE THREE NUMBERS
  • Read the question.
  • Tap your answer.
The crew's readings swing between 14 and 26. What is the amplitude of the wave?
The crew's readings swing between 14 and 26. What is the value of the midline?
The crew's readings: (0, 20), (6, 26), (12, 20), (18, 14), (24, 20). What is the period of the wave?
WRITE THE MODEL
  • Read the question.
  • Tap your answer.
The wave starts at its midline, rises to 26, falls to 14 and repeats every 24 hours. Which equation models it?
In the crew's model T(t) = 6 sin((π/12)t) + 20, what is T(6)?
The wave repeats every 24 hours. What is the period of the crew's temperature wave, in hours? Type the number.
WHY THIS EXERCISEThe period is the first of the three numbers. It sets b in the model through b = 2π ÷ period.
StatementTrue or false?
The amplitude is the distance from the low all the way to the high.?
(26 - 14) ÷ 2 = 6?
(26 + 14) ÷ 2 = 20?
The midline is the average of the highest and lowest values.?
The crew's model is a law that every greenhouse follows.?
WHY THIS EXERCISEThree numbers from the table are enough to write the whole wave. Knowing which is which is the skill.
Try it
On graph paper, plot the five readings with hours across and degrees up. Join them with a smooth curve.
Draw the dashed midline at 20. Check that the curve rises 6 above it and dips 6 below it.
Draw the temperature wave over two days. Label the amplitude, the midline and one period.

Strong start. Tomorrow you read the same three numbers from a graph and from a table, and compare the two ways.