Topic 3.7
Sinusoidal Function Context and Data Modeling
AP asks you to build a sinusoidal model from a real cycle, interpret every parameter with units, choose a phase that matches the starting event, and judge whether a prediction is reasonable.
China → AP bridge
What you already know
You likely know sine and cosine graphs, amplitude, period, and basic transformations.
What AP adds
AP asks you to build a sinusoidal model from a real cycle, interpret every parameter with units, choose a phase that matches the starting event, and judge whether a prediction is reasonable.
A domestic solution may prioritize symbolic speed. An AP solution must also select a representation, connect evidence to the requested command term, and preserve contextual units and domain.
Original worked example
Build, check, and interpret
Radius 12 gives amplitude 12; center height 18 gives midline 18; period 30 gives B=2π/30=π/15. Starting at the minimum is matched by h(t)=18−12cos(πt/15). Then h(0)=6 and h(15)=30.
半径 12 给出振幅 12，中心高度 18 给出中线 18，周期 30 给出 B=2π/30=π/15。从最低点开始，故 h(t)=18−12cos(πt/15)。检查得 h(0)=6、h(15)=30。
One model, four representations
Representation Switcher
Equivalence gate: table values are evaluated from the equation, the graph uses the same samples, and the context preserves variables and units.
AP scoring risk
Do not let a familiar habit erase evidence
Using sine by habit can shift the model by a quarter-period. Check the value and direction at t=0 before accepting the equation.
Key command terms: construct / interpret / justify