Topic 2.5
Exponential Function Context and Data Modeling
AP expects you to identify a constant ratio from data, define variables and units, construct a model, restrict the domain, and interpret growth or decay factors in context.
China → AP bridge
What you already know
You likely know exponent laws, geometric sequences, and how to solve simple exponential equations.
What AP adds
AP expects you to identify a constant ratio from data, define variables and units, construct a model, restrict the domain, and interpret growth or decay factors in context.
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
The ratios 720/960, 540/720, and 405/540 are all 0.75. With p whole shade panels, I(p)=960(0.75)^p lux. The factor 0.75 means each additional panel transmits 75% of the preceding intensity.
比值 720/960、540/720、405/540 均为 0.75。若 p 为整块遮光板数，则 I(p)=960(0.75)^p lux。因子 0.75 表示每增加一块板，透过光强是此前的 75%。
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
Calling 0.75 a “25% decrease” is useful, but an interpretation must say what retains 75% of what, per one-unit increase in the input.
Key command terms: construct / interpret / estimate