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Algebra 2 9-12 / Week 10 / Tuesday
2/6
Week 10 · Periodic Models

Tuesday

Table or graph: two ways to read a wave
// The vent timer and the temperature wave
⏱ about 20 min

Tuesday: Table or Graph, Two Ways to Read a Wave

"I read the wave from the table," Wren says. "High, low, repeat. Three lines of arithmetic."

"I read it from the picture," Comet says, tracing Nova's projected curve with a finger. "Top, bottom, one hump and one dip."

"Same three numbers?" Wren asks. "What do you notice?"

They compare. Amplitude six. Midline twenty. Period twenty-four. Both ways agree.

Nova dims the curve and shows only the dashed midline. "Would you like a hint? The midline is where the wave spends half its time above and half below."

"Now the misting bench," Comet says, pulling a second log. "Humidity. It cycles faster."

"Then the period is shorter and b is bigger," Wren says. "Let us read it both ways and see."

Way 1: from the table

  • Amplitude a = (high - low) ÷ 2.
  • Midline d = (high + low) ÷ 2.
  • Period = the x-distance from one high to the next high (or one low to the next low).
  • b = 2π ÷ period.

Way 2: from the graph

  • Draw the dashed line halfway between the top and the bottom of the wave. That is the midline, d.
  • Measure from the midline up to a peak. That is the amplitude, a.
  • Measure along x from one peak to the next. That is the period.
  • Where the wave crosses the midline going up tells you the shift c. If it crosses at x = 0, c = 0.

Both ways give the same three numbers. The table is exact. The graph shows the shape and catches a misread number fast.

The general model

y = a sin(b(x - c)) + d. The a stretches the wave up and down. The d lifts it. The b squeezes it sideways. The c slides it.

For the temperature wave, a = 6, b = π/12, c = 0 and d = 20.

Since 2π ÷ b is the period, b = π/12 means a period of 24. Then b = π/4 means a period of 8, and b = 2π a period of 1.

The misting bench log

Nova logged the humidity on the misting bench every two hours. These readings are the crew's made-up data.

HoursHumidity (%)
055
270
455
640
855
READ THE HUMIDITY WAVE
  • Read the question.
  • Tap your answer.
The humidity swings between 40% and 70%. What is the amplitude?
The humidity swings between 40% and 70%. What is the midline value?
The crew's readings: (0, 55), (2, 70), (4, 55), (6, 40), (8, 55). What is the period of the wave?
The humidity starts at its midline, rises to 70, falls to 40 and repeats every 8 hours. Which equation models it?
FROM B TO THE PERIOD
  • Read the question.
  • Tap your answer.
A wave is y = 3 sin((π/4)x) + 9. What is its period?
A wave is y = 2 sin((π/6)x) - 1. What is its period?
A wave is y = 5 sin((2π/5)x) + 7. What is its period?
In y = a sin(b(x - c)) + d, which letter is the midline? Type the letter.
Which letter is the amplitude? Type the letter.
Which letter slides the wave left or right? Type the letter.
StatementTrue or false?
Reading from the table and reading from the graph should give the same amplitude.?
(70 - 40) ÷ 2 = 15?
A bigger b means a longer period.?
The midline passes halfway between the top and the bottom of the wave.?
2 × 8 ÷ 2 = 8?
WHY THIS EXERCISETwo ways that agree are a check on each other. When they disagree, one reading is off.
Try it
On graph paper, plot the humidity table with hours across and percent up. Draw the midline at 55.
Mark the amplitude as an arrow from the midline to a peak. Mark one period along the bottom.
Draw the humidity wave and the temperature wave on the same axes, hours 0 to 24. Label each period.

Two ways, one wave. Tomorrow is Lab day: you build a wave on the bench with a jar lid and a protractor.

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