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The function y=A sin(ωx+φ)

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

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.

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.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in the function y=a sin(ωx+φ) changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

θ+φ=45° = 0.7854 rad; sin=0.7071, cos=0.7071. The circle has radius 1.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the amplitude.

y=-2\sin(2x)+3

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

Amplitude is the absolute multiplier.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$A=|-2|=2$
A negative multiplier reflects the graph but cannot make amplitude negative.

The requested value is 2.

Checks and common pitfalls: A negative multiplier reflects the graph but cannot make amplitude negative.

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

02 / Standard#Worked example

Write the least positive period as cπ; find c.

y=3\sin(2x+\pi/4)

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

Phase does not change the cycle length.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$T=2\pi/2$
The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

The requested value is 1.

Checks and common pitfalls: The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

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

03 / Transfer#Worked example

Give a positive-amplitude expression for the same graph.

y=-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

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$\sin(x+\pi)=-\sin x$
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.

04 / Foundation#Your turn

Find the amplitude.

y=-3\sin(2x)+3

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

Amplitude is the absolute multiplier.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$A=|-3|=3$
A negative multiplier reflects the graph but cannot make amplitude negative.

The requested value is 3.

Checks and common pitfalls: A negative multiplier reflects the graph but cannot make amplitude negative.

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

05 / Foundation#Your turn

Write the least positive period as cπ; find c.

y=3\sin(3x+\pi/4)

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

Phase does not change the cycle length.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$T=2\pi/3$
The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

The requested value is 0.66666667.

Checks and common pitfalls: The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

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

06 / Foundation#Your turn

State the horizontal shift of this graph relative to y=sin(3x).

y=\sin(3x+\pi/2)

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

Factor out the frequency before reading the shift.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$3x+\pi/2=3(x+\pi/6)$
The horizontal shift is phase divided by frequency, with opposite sign.

Left by π/6.

Checks and common pitfalls: The horizontal shift is phase divided by frequency, with opposite sign.

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.

07 / Foundation#Your turn

Find the value at x=0.

y=3\sin(2x+\pi/2)+1

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

Substitute into the whole internal angle.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$y(0)=3\sin(\pi/2)+1=4$
The vertical shift is added after multiplying sine.

The requested value is 4.

Checks and common pitfalls: The vertical shift is added after multiplying sine.

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

08 / Standard#Your turn

Find A>0 from the maximum and minimum.

y=A\sin x+c,\quad y_{\max}=7,\quad y_{\min}=1

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

Amplitude is half the peak-to-trough distance.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$A=\frac{(7)-1}2=3$
The midline is the average of the two extreme values.

The requested value is 3.

Checks and common pitfalls: The midline is the average of the two extreme values.

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

09 / Standard#Your turn

What function remains when the frequency is zero? Does it have a least positive period?

y=3\sin(0x+\pi/2)

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

Every positive shift preserves a constant function.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$y=3;\quad f(x+T)=f(x)\ (T>0)$
Because arbitrarily small positive shifts work, there is no least one.

The constant 3; it has no least positive period.

Checks and common pitfalls: Because arbitrarily small positive shifts work, there is no least one.

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.

10 / Standard#Your turn

Give a positive-amplitude expression for the same graph.

y=-3\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

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$\sin(x+\pi)=-\sin x$
Different parameters can describe the same graph.

y=3sin(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.

11 / Standard#Your turn

Find the amplitude.

y=-4\sin(2x)+3

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

Amplitude is the absolute multiplier.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$A=|-4|=4$
A negative multiplier reflects the graph but cannot make amplitude negative.

The requested value is 4.

Checks and common pitfalls: A negative multiplier reflects the graph but cannot make amplitude negative.

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

12 / Transfer#Your turn

Write the least positive period as cπ; find c.

y=3\sin(4x+\pi/4)

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

Phase does not change the cycle length.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$T=2\pi/4$
The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

The requested value is 0.5.

Checks and common pitfalls: The nonzero amplitude and frequency guarantee a nonconstant sinusoid.

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

13 / Transfer#Your turn

State the horizontal shift of this graph relative to y=sin(4x).

y=\sin(4x+\pi/2)

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

Factor out the frequency before reading the shift.

Worked solution

Separate amplitude, internal frequency, phase and vertical shift.
Calculate or simplify this relation.
$4x+\pi/2=4(x+\pi/8)$
The horizontal shift is phase divided by frequency, with opposite sign.

Left by π/8.

Checks and common pitfalls: The horizontal shift is phase divided by frequency, with opposite sign.

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.

End-of-lesson check and correction

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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.