Non-linear Relationships - Algebra-1

1. Fundamental Concepts

A non-linear relationship refers to a relationship between variables that is not in the form of a straight line, and its function graph is not a straight line. Unlike linear relationships where the rate of change of variables is constant, in non-linear relationships, the rate of change of variables is not fixed. It does not conform to the form of a linear function (i.e., \(y = kx + b\), where k and b are constants and \(k≠0\)).

Methods to identify non-linear relationships: This is the key to determining whether a relationship is non-linear, distinguishing non-linear relationships from linear ones through specific characteristics and criteria.
Common \(f(x)=x^n\) relationships: These are a typical type of non-linear relationships, where n is a constant and \(n≠1\). The properties and graphs of such functions have their unique characteristics.
Understanding from the perspectives of graphs and tables: In terms of graphs, the graph of a non-linear relationship is not a straight line but may be in the form of a curve or other shapes; in tables, the changes between variables do not show a constant ratio or difference.

\(f(x)=x^2\), which is the most basic quadratic function. It belongs to the case of \(f(x)=x^n\) when \(n = 2\), and its graph is a parabola opening upwards.
\(f(x)=x^3\), which is a cubic function. Its graph is an S-shaped curve and is monotonically increasing within its domain.
\(f(x)=x^4 - 2x^2 + 1\), this function is a quartic function. Its graph is more complex, with multiple extreme points and inflection points.

Judgment from graphs: Observe the graph of a function. If the graph is not a straight line, the function may represent a non-linear relationship. For example, graphs such as parabolas and curves usually correspond to non-linear relationships.
Analysis from tables: Check the changes of variables in the table. If the independent variable increases by the same amount each time, but the change amount of the dependent variable is not constant, it may be a non-linear relationship.
Judgment based on function expressions: If the degree of the variable in the function expression is not 1, such as containing quadratic, cubic, or higher - degree terms, or the variable is in the denominator, under the square root, etc., it is usually a non-linear relationship.