Kenneth ChengMathematical modeling
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Give a positive-amplitude expression for the same graph.

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: Trigonometric identities and transformations

TOPIC 01

The function y=A sin(ωx+φ)

Build understanding of the function y=a sin(ωx+φ) through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use the function y=a sin(ωx+φ) with explicit angle units and domains.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

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

01 / Transfer#Worked example

Give a positive-amplitude expression for the same graph.

y=−2sin⁡xy=-2\sin x
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Separate amplitude, internal frequency, phase and vertical shift.
Hint 2
A half-cycle shift reverses sine.
Worked solution

  1. Separate amplitude, internal frequency, phase and vertical shift.

  2. Calculate or simplify this relation.

    sin⁡(x+π)=−sin⁡x\sin(x+\pi)=-\sin x

  3. Different parameters can describe the same graph.

y=2sin(x+π).

Checks and common pitfalls: Different parameters can describe the same graph.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

Focus on one question

Teacher preparation and assessment

Question sequence

  • What must be true before using the main rule for the function y=a sin(ωx+φ)?
  • Which representation makes this task easier, and why?
  • Change one assumption. Does the conclusion survive?

Board plan

  • Use the function y=a sin(ωx+φ) with explicit angle units and domains.
  • Scale and cycle: Amplitude is |A|; for a nonconstant graph the least positive period is 2π/|ω|.
  • Phase shift: Factor ω to read the horizontal shift −φ/ω.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: Amplitude is |A|; for a nonconstant graph the least positive period is 2π/|ω|.
  • Expected reasoning: Factor ω to read the horizontal shift −φ/ω.
  • Expected correction: A constant graph has no least positive period.

Assessment checklist

  • 1: identify the givens and required quantity.
  • 1: choose a valid definition, representation or method.
  • 1: present connected, correct reasoning.
  • 1: check conditions and explain the result.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗