Topic 1.11
Equivalent Representations
AP asks which form makes a feature visible: zeros in factored form, end behavior in expanded form, and values or sign changes in tables and graphs.
China → AP bridge
What you already know
You likely know expansion, common factors, the difference of squares, and solving polynomial equations.
What AP adds
AP asks which form makes a feature visible: zeros in factored form, end behavior in expanded form, and values or sign changes in tables and graphs.
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
For P(x)=x³−4x, factor to P(x)=x(x−2)(x+2). The three real zeros are −2, 0, and 2. Because the leading term is x³, the left tail falls and the right tail rises.
对 P(x)=x³−4x，分解为 P(x)=x(x−2)(x+2)。三个实零点为 −2、0、2。首项为 x³，所以左端下降、右端上升。
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
Expanding automatically can hide the very feature the prompt asks you to determine. State the form and the feature it reveals.
Key command terms: determine / construct / justify