Study guide: Interpret and Construct Periodic Models
Read, build, shift, and solve levels of supplied cosine functions over stated intervals.
Ideas to explain in your own words
- Identify a midline, amplitude and period in a supplied periodic function.
- Construct a cosine model from extrema, period, and starting phase.
- Use a horizontal shift to locate peaks and quarter-period values.
- Solve a cosine-level equation over one stated cycle.
Watch your reasoning
- At t = 6 the angle is π and its cosine is −1, so k(6) = 5 − 2 = 3. Reducing the amplitude changes the range to 3 through 7 while the period stays 12 hours.
- The midline is (9 + 1)/2 = 5, amplitude (9 − 1)/2 = 4, and angle coefficient 2π/12 = π/6. The first rule has all three features and the required phase.
- At t = 4, t − 4 = 0 and cosine is one, giving the maximum 8. The preceding peak would be t = −8, outside nonnegative time.
- A midline value requires cosine zero. The supplied angles π/2 and 3π/2 give t = 3 and t = 9. Both lie in 0 ≤ t < 12.
A worked review example
Read a cosine model: h(0) = 5 + 3 = 8. h(3) = 5 + 3 × 0 = 5. h(6) = 5 + 3 × (−1) = 2. h(12) = 8 again, consistent with a 12-hour period. Build a Cosine Rule from Features: h(t) = 7 + 3cos(πt/4). At t = 0, h = 7 + 3 = 10; at t = 4, h = 7 + 3(−1) = 4. Increasing t by 8 adds 2π to the angle. Track a Shifted Cosine Pattern: h(2)=8, h(5)=5, h(8)=2, and h(14)=8. The angles are respectively 0, π/2, π, and 2π, so the peak-to-peak interval is 14 − 2 = 12 hours. Find Two Times at a Given Level: πt/6 = π/3 gives t = 2; πt/6 = 5π/3 gives t = 10. Both yield 5 + 3(1/2) = 6.5, and both are in the stated cycle.
Make a practice plan
- Try the supplied lesson problems without looking at the answers.
- Compare your method with the worked examples and explain any correction.
- Use the new review cases, then complete the transfer task on paper or in your own document.