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What is the horizontal shift?

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

Return to the lesson / paper ↗

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

Revisit first: Unit-circle coordinates and signs Identity choice and domain preservation

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Read the displacement	Use the phase diagnostic and require factoring before choosing.
8	Separate parameters	Annotate amplitude, midline and one full cycle on a graph.
10	Reconstruct and qualify	Pairs reconstruct from extrema and discuss why phase remains free.
12	Interpret and sample	Groups solve the tide threshold and devise a measurement to distinguish sampled models.
5	Exit direction	Distinguish 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.
$\omega(x+\phi/\omega)$
Retain time units in a model.
$T=12\text{ hours}$

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.