KKhaemenes Algebra IIOpen-Age Algebra II & Advanced Functions
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Unit 10 · Lesson 06 · Open-Age

Amplitude, Period, Phase & Vertical Shifts

A controls vertical scale, B controls period, C phase shift, and D midline. Context determines which features have physical meaning.

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Learning objectives

  • Explain the central structure of amplitude, period, phase & vertical shifts using precise mathematical language.
  • Apply the governing relationship and verify results with an alternate representation or substitution.
  • State domain, unit, precision, and context restrictions whenever they affect the result.
  • Analyze a misconception and communicate a correction with evidence.

Essential question: How do angle measure, the unit circle, graphs, and periodic functions model rotation, waves, cycles, and repeated behavior?

Governing relationship

y=A sin(B(x−C))+D; amplitude |A|; period 2π/|B|.

A controls vertical scale, B controls period, C phase shift, and D midline. Context determines which features have physical meaning.

Worked examples

Read each example, close it, and reconstruct the reasoning independently. Then verify by substitution, graph, table, or alternate method.

Analyze y=3sin(2x−π)+4.Rewrite 2(x−π/2): amplitude 3, period π, shift right π/2, midline 4.
Build a model with max 10, min 2, period 6.Amplitude 4, midline 6, B=2π/6=π/3.

Reasoning workshop · Factor first to identify phase shift

Study the governing rule, follow every step, and verify the conclusion before attempting independent work.

Investigation A · Analyze y=3sin(2x−π)+4.

  1. Factor the inside: 2x−π=2(x−π/2), so the shift is right π/2.
  2. Amplitude is 3, period 2π/2=π, and midline is y=4.
  3. The range is [1,7]. At x=π/2 the inner argument is 0, so y=4.

Verification: At x=3π/4, the inner argument is π/2 and the maximum y=7 occurs.

Investigation B · Model a cycle with minimum 3, maximum 7 and period 6.

  1. Midline (7+3)/2=5; amplitude (7−3)/2=2.
  2. For period 6, the angular frequency is B=2π/6=π/3.
  3. To make the maximum occur at t=1, choose y=5+2cos((π/3)(t−1)).

Verification: At t=1, y=7; at t=4, y=3, exactly half a period later.

Concept diagram · Factor first to identify phase shift

Factor first to identify phase shift3sin(2x−π)+4 then 3sin(2(x−π/2))+4 then Amplitude 3; period π; midline 43sin(2x−π)+4Given3sin(2(x−π/2))+4ReasonAmplitude 3; period π; midline 4Check
A diagram of the mathematical reasoning sequence: 3sin(2x−π)+4 → 3sin(2(x−π/2))+4 → Amplitude 3; period π; midline 4. Explain how the result is independently verified.

Independent transfer · check after solving

Complete these without the game or answer key first. Give exact values where possible and show a short verification.

  1. Describe amplitude, period, shift, midline and range of y=−2sin(4(x−π/8))+1.
  2. Evaluate that function at x=π/8.
  3. Build a cosine model with max 10, min 2 and period 6, peaking at 0.
Reveal independent answer reasoning
  1. Amplitude 2; period π/2; right shift π/8; midline 1; range [−1,3], with reflection.
  2. The sine argument is 0 and y=1.
  3. One model is y=6+4cos(πx/3).

Misconception to correct: In 2x−π, π is not the shift. Factor the coefficient 2 to find the actual rightward shift π/2.

Compare your solution to the verified reasoning, correct one mistaken step, and then use optional game practice. Official grading stays separate.

Guided and independent practice

  1. Find amplitude and period.
  2. Write a sinusoid from extrema.
  3. Determine phase shift carefully.
  4. Create one original example involving amplitude, period, phase & vertical shifts and solve it completely.
  5. Use the Khaemenes Scientific Calculator to verify one result, then document the setup and reasonableness check.
  6. Write a one-paragraph explanation connecting symbolic, graphical, numerical, and contextual representations.

Differentiated pathways

Foundation

Review prerequisite vocabulary, annotate each worked step, complete the first four prompts, and use a table or graph to support symbolic work.

Core

Complete every prompt, explain domain and verification, correct errors, and submit the applied task with clear mathematical communication.

Extended

Generalize the relationship, compare methods, use technology strategically, analyze sensitivity, and create a proof, counterexample, or advanced model.

Application laboratory

Build a sinusoidal model from cycle length, maximum, minimum, and timing.

Required evidence: defined quantities and units; at least two representations; transparent calculations; verification; one limitation; and a defensible conclusion.

Misconception and correction

Common misconception: Reading C directly before factoring B from the angle expression.

Write the first invalid step, the governing rule that corrects it, and one counterexample or verification that demonstrates the correction.

Mastery evidence

  • Accurate symbolic work with preserved equivalence.
  • Correct domain, interval, unit, and precision statements.
  • At least one independent verification.
  • A correction explaining why an incorrect approach fails.
  • A clear application conclusion that distinguishes mathematical result from recommendation.

Reflection

What structure did you recognize? Which representation was most useful? What assumption or restriction mattered most? What will you do differently on the next problem?