Identify linear and nonlinear functions

Learning Objectives Define linear and nonlinear functions. Identify linear functions from their equations. Identify nonlinear functions from their equations. Determine if a function represented by a table of values is linear or nonlinear. Distinguish between linear and nonlinear functions when represented graphically. Explain the key characteristics that differentiate linear from nonlinear functions. Ever notice how some things grow steadily, while others explode or slow down? 📈 Understanding these different patterns helps us predict the future! In this lesson, you'll learn to recognize two fundamental types of relationships: linear and nonlinear functions. We'll explore how to identify them from equations, tables, and graphs, which is a crucial skill for underst...

TermDefinitionExample FunctionA relationship where each input (x-value) has exactly one output (y-value).The relationship between the number of hours worked and the money earned (if paid hourly). Linear FunctionA function whose graph is a straight line and has a constant rate of change.The equation $y = 2x + 3$ represents a linear function. For every increase of 1 in x, y increases by 2. Nonlinear FunctionA function whose graph is not a straight line and has a varying rate of change.The equation $y = x^2$ represents a nonlinear function. As x increases, y increases at an accelerating rate. Rate of ChangeA measure of how much the output (y) changes for each unit change in the input (x).If a car travels 60 miles in 1 hour, its rate of change (speed) is 60 miles per hour. SlopeThe steepness...

Standard Form of a Linear Function $y = mx + b$ An equation represents a linear function if it can be written in this form, where $m$ is the constant slope (rate of change) and $b$ is the y-intercept. The variable $x$ must have an exponent of 1 (even if not explicitly written), and there should be no variables multiplied together or in denominators/roots. Constant Rate of Change (Slope Formula) $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$ To check if a function represented by a table of values is linear, calculate the rate of change between several pairs of points. If the rate of change ($m$) is constant for all pairs, the function is linear. Characteristics of Nonlinear Functions from Equations An equation is nonlinear if the variable $x$ has an expon...