Unit 05 · Lesson 01 · Week 13 · Trigonometric Functions, Graphs & Periodic Models

Graphing Sine and Cosine from the Unit Circle

Interpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning. In this lesson, the central focus is graphing sine and cosine from the unit circle. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.

Open-AgePrintableFoundation · Core · ExtendedMastery target 80%

Learning intentions

Interpret the defining quantities and notation used in graphing sine and cosine from the unit circle.
Construct or analyze at least two representations related to graphing sine and cosine from the unit circle.
Show symbolic work with correct domains, units, intervals, or angle measures.
Explain why the selected method is valid and check the result against another representation.

Essential question

How do amplitude, period, phase, midline, and reciprocal behavior describe and model repeating phenomena?

Multiple-representation model

Interpret and construct periodic functions by tracking amplitude, period, frequency, phase, midline, asymptotes, and contextual meaning.

y=A\sin(B(x-C))+DT=\frac{2\pi}{|B|}y=A\cos(B(x-C))+D

Worked reasoning

Parameter readingFor y=3 sin(2(x−π/4))+1, amplitude is 3, period is π, phase shift is π/4 right, and midline is y=1.

Model defenseIdentify max, min, and one full cycle first; then verify that the proposed equation reaches the observed extrema at the correct input values.

Misconception check: The coefficient B changes the period reciprocally; the period is not simply B.

Communication and reasoning

Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.

Khaemenes Academy · Open-Age Precalculus

Graphing Sine and Cosine from the Unit Circle

Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.

U05 · L01

Name

Date

For y=4sin(3x)−2, state the amplitude.

For y=4sin(3x)−2, state the period.

For y=2cos(x−π/4)+1, state the phase shift.

State the midline of y=−3sin(2x)+5.

State one vertical asymptote of y=tan x.

A periodic quantity ranges from 8 to 20. Find amplitude and midline.