Linear VS. Nonlinear - Algebra-1

1. Fundamental Concepts

Definition: Linear relationships are characterized by equations of the form $$y = mx + b$$, where $$m$$ and $$b$$ are constants, and the graph is a straight line.
Nonlinear Relationships: Nonlinear relationships cannot be expressed in the form $$y = mx + b$$ and their graphs are not straight lines. Examples include quadratic, exponential, and trigonometric functions.
Key Differences: Linear relationships have a constant rate of change, while nonlinear relationships do not.

Linear Equation Form: $$y = mx + b$$

Nonlinear Example: $$y = x^2 + 3x - 2$$

Rate of Change: In linear relationships, the rate of change is constant; in nonlinear, it varies.

Problem: Identify if the equation $$y = 4x + 7$$ is linear or nonlinear.

Validation: The equation $$y = 4x + 7$$ represents a straight line on a graph, confirming it is linear.

Problem: Determine if the equation $$y = x^3 - 2x + 5$$ is linear or nonlinear.

The highest power of $$x$$ is 3, which means it does not fit the form $$y = mx + b$$.
Therefore, it is a nonlinear relationship.

Validation: Graphing $$y = x^3 - 2x + 5$$ shows a curve, confirming it is nonlinear.

Graphical Method: Plot the equation to visually determine if the graph is a straight line (linear) or a curve (nonlinear).
Algebraic Analysis: Check the highest power of the variable. If it is 1, the relationship is linear; otherwise, it is nonlinear.
Table of Values: Create a table of values for the equation and observe if the rate of change between consecutive points is constant (linear) or varies (nonlinear).