Nonlinear functions
Nonlinear functions on the SAT center on quadratics and exponentials — recognizing their shapes, connecting equations to graphs, and interpreting features like vertex, intercepts, and growth/decay in real-world contexts.
核心知识
Quadratics produce symmetric parabolas with a single turning point; exponentials produce curves that grow (or decay) by a constant ratio each step. Recognizing which pattern fits — and quickly extracting key features from the equation form you're given — is the core skill the SAT tests.
深入理解
Most nonlinear-function questions fall into a few buckets:
Model selection: Does the scenario describe constant differences (linear), constant second differences (quadratic), or constant ratios (exponential)?
Equation ↔ graph: Each quadratic form (standard, vertex, factored) reveals different features directly. Exponential equations
𝑓 ( 𝑥 ) = 𝑎 ⋅ 𝑏 𝑥 𝑎 𝑏 Interpretation: The SAT often asks what a specific number in an equation means in the situation described. Vertex coordinates become maximum height or optimal time; intercepts become starting values or break-even points; the base of an exponential becomes a percent increase or decrease.
Transformations: Shifting, stretching, compressing, and reflecting graphs follow consistent algebraic rules that apply to any function type.
知识点教程
8Decide whether a quadratic or exponential model fits a context or dataset.
Choosing between quadratic and exponential models based on how data changes.
Build or use a quadratic model; interpret vertex and intercepts in context.
Setting up quadratic equations from word problems and reading meaning from their vertex and intercep
Build or use an exponential model; interpret initial value and growth/decay factor in context.
Working with exponential equations and understanding what each part means in a real scenario.
Identify and interpret key graph features (intercepts, vertex/extrema, maximum/minimum).
Reading intercepts, vertex, and extrema from graphs and explaining what they mean.
Determine key features (e.g., y-intercept, vertex) from a nonlinear graph or equation.
Extracting intercepts, vertex, and other features algebraically from equations.
Connect equation forms to graph features (factored/standard/vertex form for quadratics).
Matching the three quadratic forms to the graph features each one reveals most directly.
Interpret parameters, constants, and input–output pairs in context.
Explaining what numbers and variables in an equation mean within the real-world situation described.
Apply transformations (shift, stretch/compress, reflection) to nonlinear graphs and equations.
Moving, stretching, compressing, and flipping graphs using algebraic changes to the equation.