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For a constant function, what is the least positive period?

Read the idea, work independently, then explain what changed.

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高一必修 第一册(A版).pdf · 5.6 · PDF 238 / printed page 231

Revisit first: Unit-circle coordinates and signsIdentity choice and domain preservation

TOPIC 01

Amplitude, period and phase in models

Read parameters as observable changes and reconstruct models without hiding ambiguity.

What you will be able to explain

  • Distinguish amplitude, vertical shift, angular frequency and horizontal shift.
  • Reconstruct a sinusoid from sufficient observations and recognise when samples are insufficient.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Your turn

For a constant function, what is the least positive period?

y=4y=4
  • All inputs are real unless a smaller domain is stated.
Skills and prerequisite lessons

Scope: JM01 / JM02

  1. 2π
  2. 0
  3. 4
  4. No least positive period
Working and explanation
BUILD THE REASONING
Hint 1
Every positive shift works.
Hint 2
For any proposed positive period, half of it also works.
Worked solution
  1. Compare shifted outputs.
    f(x+T)=4=f(x)f(x+T)=4=f(x)
  2. There is no smallest positive choice.
    T/2<T(T>0)T/2<T\quad(T>0)
D: none exists.
Checks and common pitfalls: Zero is not a positive period, and choosing 2π is not least.
Think first. Reveal a hint when the class is ready.
Focus on one question
Teacher preparation and assessment
40-minute lesson
MinutesFocusTeaching and student action
5Read the displacementUse the phase diagnostic and require factoring before choosing.
8Separate parametersAnnotate amplitude, midline and one full cycle on a graph.
10Reconstruct and qualifyPairs reconstruct from extrema and discuss why phase remains free.
12Interpret and sampleGroups solve the tide threshold and devise a measurement to distinguish sampled models.
5Exit directionDistinguish an upward crossing from an arbitrary zero.
Question sequence
  • Which measured feature fixes each parameter?
  • Could two different continuous curves fit all samples?
Board development
  • Factor before reading horizontal shift.
    ω(x+ϕ/ω)\omega(x+\phi/\omega)
  • Retain time units in a model.
    T=12 hoursT=12\text{ hours}
Expected correct and incorrect responses
  • “Range determines A,D but not phase” is a correct limitation.
  • “All readings equal, so the function is constant” overclaims the data.
Follow-up questions
  • What observation fixes the direction of a crossing?
Remediation
  • For phase errors, solve ωx+φ=0 directly, then compare with the claimed shift.
Exit check
  • For y=−3sin(2x)+1, give amplitude, range and least period. Expected 3, [−2,4], π.
Assessment checklist and sample responses
  • 2 marks: parameter roles.
    Full: divide phase by frequency; partial: use phase alone as shift.
  • 2 marks: evidence and ambiguity.
    Full: two fitted models plus t=1/4 distinguishing test; partial: two formulas without checking samples.
No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.
Curriculum and source notes ↗