Nonlinear functions form a critical component of the SAT math section. This guide covers everything from quadratic to absolute value functions, offering strategies and examples to master these concepts.
Understanding nonlinear functions is an essential skill for any student preparing for the SAT math section. Unlike linear functions, nonlinear functions introduce many more complexities and strange-looking graphs.
By breaking down these functions into their core components, this guide introduces the various forms of nonlinear functions that can show up on the SAT exam and shows students how to solve questions on each type.
Nonlinear functions have variable rates of change, creating curves rather than straight lines. For the SAT, focus on these 4 types:
Quadratic Functions: Represented as ax^2 + bx + c, where a is not 0. They form parabolas on a graph.
Higher-Order Polynomial Functions: Include terms with variables raised to powers higher than two, such as ax^3 + bx^2 + cx + d.
Exponential Functions: Have the form a * b^x, where the variable x is in the exponent, leading to rapid increases or decreases.
Absolute Value Functions: Characterized by the absolute value symbol, producing a V-shaped graph, as in |x|.
Quadratic functions are represented by ax^2 + bx + c. The graph is a parabola that opens upward (a > 0) or downward (a < 0). They appear in standard, factored, and vertex forms on the SAT.
Mastering factoring, completing the square, and the quadratic formula is essential for success.
These feature variables raised to powers greater than two: cubic (third degree), quartic (fourth degree), etc. A general form is ax^n + bx^(n-1) + ... + k.
Solving requires factoring techniques, and sometimes finding a root by inspection first.
Exponential functions have the form a * b^x and represent rapid growth or decay. The variable is in the exponent, leading to faster-than-linear change.
Common applications: population growth, radioactive decay, compound interest.
Absolute value functions have the form y = |ax + b| + c and produce V-shaped graphs. They turn all negative inputs into positive outputs.
Solving involves splitting into two cases: one where the expression is positive, and one where it is negative.
Understand Basic Graph Shapes: Recognize parabolas for quadratics, exponential curves, and V-shapes for absolute value. This helps quickly identify function types.
Practice Factoring Techniques: Mastery in factoring polynomials is key. Practice grouping, the quadratic formula, and synthetic division.
Learn Exponent and Logarithm Properties: Understanding these allows easier equation simplification and variable isolation.
Use Substitution for Complex Equations: Replace part of the equation with a simpler variable, solve, then back-substitute.
Sketch Quick Graphs: A quick graph provides insights into roots, symmetry, and intervals of increase or decrease.