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Grade 12 · unit review

Review: Interpret and Construct Periodic Models

Read, build, shift, and solve levels of supplied cosine functions over stated intervals.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. For h(t)=6+4cos(πt/4), use cos(π)=−1. What is h(4)?
2. A cosine signal starts at a maximum of 14, has minimum 6 and period 10. Which rule fits?
3. For q(t)=7+2cos(π(t−1)/4), when is the first nonnegative maximum?
4. For p(t)=2+4cos(πt/3), 0 ≤ t < 6, when is p(t)=4? Use cos(θ)=1/2 at π/3 and 5π/3 only in one cycle.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Prepare a fictional sensor-model note. The signal has maximum 12 units, minimum 4, period 8 hours, and its first nonnegative peak at t = 1. Build a positive-amplitude cosine rule, state its midline, amplitude, range, and period, and find its first minimum. In 1 ≤ t < 9, find both times it equals 10. Use cos(θ)=1/2 at π/3 and 5π/3 only in one cycle. These are supplied model values, not a prediction about a real sensor.

Work on paper or in your own document. Explain the choices you made.

  • Find midline 8, amplitude 4, range 4 through 12, and period 8.
  • Construct h(t)=8+4cos(π(t−1)/4).
  • Locate the first minimum at t = 5 and value 4.
  • Solve h(t)=10 to get t = 7/3 and t = 23/3 within 1 ≤ t < 9.
  • Check both angles and explain that all results describe the supplied fictional model.
Compare with a model response

Midline = (12 + 4)/2 = 8 and amplitude = (12 − 4)/2 = 4. The rule is h(t)=8+4cos(π(t−1)/4), with range 4 to 12 and period 8. Its minimum is at t = 1 + 4 = 5, giving 4. For level 10, cosine must be 1/2. Thus t − 1 = 4/3 or 20/3, giving t = 7/3 and 23/3; both fall in 1 ≤ t < 9.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.