Mr. S Teaches
Sign inRequest Tutoring
Menu
Sign inRequest Tutoring

Precalculus and Trigonometry · 35 minutes

Suggested grades 11–12 · Unit 3: Periodic models and finite sequences · Lesson 1 of 2

A Sinusoidal Wheel-Height Model

Choose amplitude, midline, period and phase from a fictional rotating wheel.

Mr. S Teaches · Original starter lesson · HS-PRE-05 · Grade range is planning guidance.

Watch, notice, discuss

Video choices for this lesson

2 choices

Preview a video, then choose what fits your learners. “Show to class” opens a larger view; use the player’s play and full-screen controls when you’re ready.

Teacher preview needed. Check the language, level and length before showing. YouTube may show ads or related videos.

After watching

How do the wheel’s radius, center height, period and starting position appear in the model?

An external player loads only after you choose Preview or Show to class. Playback connects to YouTube. Videos belong to their creators; availability and embedding can change.

Learning objective

Construct and check a periodic model at key quarter-period times.

Before you begin

  • Unit-circle sine and cosine
  • Function transformations

Gather your materials

  • Paper and pencil
  • Graph paper
  • Calculator for checking results

Language to know

amplitude
The maximum vertical distance from a periodic model’s midline.
period
The input interval for one full repeating cycle.
midline
The horizontal level halfway between maximum and minimum outputs.

Understand the concept

A fictional wheel’s seat ranges from 2 to 18 m and completes a turn in 40 s. Its midline is 10 and amplitude 8. A seat beginning at the lowest point is conveniently modeled by h(t)=10−8 cos(2πt/40). The model assumes steady rotation and ideal circular motion; it is a mathematical scenario, not an operating instruction.

Teach it step by step

  1. Calculate midline(18+2)/2 and amplitude(18−2)/2.
  2. Convert one turn per 40 seconds into angular factor 2π/40.
  3. Choose a negative cosine because h(0) should equal the minimum.
  4. Evaluate at 0,10,20,30,40 seconds and sketch one cycle with units.

Make the thinking visible

Worked example or model response

h(t)=10−8 cos(πt/20) gives heights 2,10,18,10,2 m at the listed times. Changing only the period to 20 s doubles the frequency, not the amplitude.

Guided practice

Build a model with minimum 3, maximum 15, period 24 and a start at the highest point.

Independent practice

1. State amplitude, midline and period of y=4+3 sin(πt/6). 2. Explain why a model starting at its midline may need a different phase from the seat model.

Quick assessment

What check immediately detects the wrong sign on the model’s cosine term?

Answer key & assessment criteria

Answers and teaching guidance

  1. Guided: h(t)=9+6 cos(πt/12); alternatives with equivalent phase are acceptable.
  2. Independent 1: Amplitude 3, midline 4, period 12. 2: At t=0, a sine-based form gives a midline crossing, whereas the seat starts at an extremum.
  3. Quick check: Evaluate t=0; the seat must begin at 2 rather than 18.

Look for: Require the learner to explain the interpretation and check, not only state a result. Distinguish an exact value from a rounded approximation and attend to the stated domain.

Make it work for your learner

More support

Use a five-point quarter-cycle table before writing the full function.

Extension challenge

Model the same seat with a sine phase shift and verify the key times.

Verified references · Selected contributions

Standards connections

Subject-content standards

These records identify the part of a subject standard practiced here. Course placement is instructional guidance; the source’s category or band remains authoritative.

Contributes to Common Core standard

HSF.TF.B.5

Common Core · High school · Functions

Regular high school mathematics standard

Choose a trigonometric model with specified amplitude, frequency and midline.

Read the official standard ↗

This lesson contributes selected skills; broader applications and full standard coverage require further instruction.

Verified 2026-11-04. Selected contribution, not full attainment.