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Algebra 2 9-12 / Week 10 / Thursday
4/6
Week 10 · Periodic Models

Thursday

Shift it, stretch it, squeeze it
// The vent timer and the temperature wave
⏱ about 20 min

Thursday: Shift It, Stretch It, Squeeze It

"Suppose next week runs three degrees warmer all day," Comet says. "Same vents, same timer."

"Then every reading goes up by three," Wren says. "The whole wave lifts. Midline 23, same amplitude."

"And if we open the vents wider, so the swing doubles?"

"Amplitude 12. The wave stretches up and down from the midline. What do you notice about the period?"

"Unchanged," Comet says. "Now the fun one. The timer runs six hours late."

Nova slides the projected wave to the right. "Would you like a hint? The peak moves, the shape does not."

"T(t - 6)," Wren says. "The peak lands at hour 12 instead of 6. Four changes, four effects. Let us prove the period rule too."

Four changes to one wave

Start with the midline-zero wave W(t) = 6 sin((π/12)t), the temperature above or below the midline. Its peak is 6 at hour 6.

New ruleWhat changesEffect on the graph
W(t) + kevery output rises by kslides up k (the midline moves)
k × W(t)every output is multiplied by kstretches up and down (amplitude times k)
W(t - k)the input is used k laterslides right k (the peak moves later)
W(k × t)the input runs k times fastersqueezes sideways (period divided by k)
The temperature wave slid 6 hours to the right: the same shape, the peak now at hour 12.
The temperature wave with its amplitude doubled to 12: the same midline and period, twice the swing.

A negative k runs each effect the other way: W(t) - 3 slides down, and W(t + 6) slides left.

Why the period is 2π ÷ b

  1. The sine function repeats when its input grows by 2π: sin(u + 2π) = sin u.
  2. In sin(bx) the input is u = bx.
  3. When x grows by some amount p, the input grows by b × p.
  4. The wave repeats as soon as b × p = 2π.
  5. Solve for p: p = 2π ÷ b. That is the period.

Check: b = π/12 gives p = 2π ÷ (π/12) = 24. The crew's wave repeats every 24 hours, as the log showed.

THE PERIOD PROOF, IN ORDER
  • ?So p = 2π ÷ b is the period
  • ?In sin(bx) the input is bx
  • ?When x grows by p, the input grows by b × p
  • ?The sine function repeats when its input grows by 2π
  • ?The wave repeats when b × p = 2π
WHY THIS EXERCISEThe rule b = 2π ÷ period is not a trick to memorize. It falls straight out of what sine does.
The x-distance before a wave repeats is its what? Type one word.
W(t) + 3 moves the wave up. Which of the three numbers changes? Type one word.
2 × W(t) doubles the swing. Which number doubles? Type one word.
READ EACH EFFECT
  • Read the question.
  • Tap your answer.
W(t) = 6 sin((π/12)t). Let g(t) = W(t) + 3. What is the highest value of g?
Let g(t) = 2 × W(t). What is the amplitude of g?
Let g(t) = W(t - 6). At what hour is g first at its peak?
Let g(t) = W(2t). What is the period of g, in hours?
WHICH MODEL MAKES SENSE?
  • Read the question.
  • Tap your answer.
A warmer week lifts every reading by 3 but keeps the same swing. Which rule fits?
The timer is reset so the peak comes 6 hours earlier. Which rule fits?
StatementTrue or false?
Multiplying the output by 2 doubles the period.?
W(t - 6) slides the wave 6 units to the right.?
2 × 12 = 24?
Adding 3 to the output changes the midline but not the amplitude.?
6 × 2 = 12?
WHY THIS EXERCISEOutput changes and input changes act in different directions. That one idea sorts all four rules.
Draw W(t) and 2 × W(t) on one set of axes, then W(t) and W(t - 6) on another. Label each change.

Sharp reasoning. Tomorrow you read key features off the wave and use the model to fill in the hours between readings.

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