Identify linear and nonlinear functions

Learning Objectives Define a function and differentiate between its input and output. Identify linear functions from their equations, recognizing the form $y = mx + b$. Determine if a function represented by a table of values is linear or nonlinear by examining its rate of change. Distinguish between linear and nonlinear functions by analyzing their graphs. Classify real-world scenarios as representing linear or nonlinear relationships. Explain the key characteristics that define a linear function. Have you ever noticed how some things grow steadily, like the cost of apples per pound, while others grow super fast or slow down, like the spread of a rumor? 🍎💨 In this lesson, you'll learn to identify different types of relationships between quantities, specifically focu...

TermDefinitionExample FunctionA relationship where each input has exactly one output.For the function $y = 2x$, if the input is 3, the output is always 6. Input (Independent Variable)The value that is put into a function, often represented by 'x'. It's the variable that changes independently.In the equation $y = 5x + 10$, 'x' is the input. If x represents hours worked, it's the input. Output (Dependent Variable)The value that comes out of a function, often represented by 'y'. Its value depends on the input.In the equation $y = 5x + 10$, 'y' is the output. If y represents total earnings, it's the output. Linear FunctionA function whose graph is a straight line and has a constant rate of change. Its equation can be written in the form $...

Slope-Intercept Form of a Linear Equation $y = mx + b$ This is the most common form for a linear equation. 'm' represents the constant rate of change (slope), and 'b' represents the y-intercept (the value of y when x is 0). If an equation can be rearranged into this form, it's linear. Slope Formula (Rate of Change) $m = \frac{y_2 - y_1}{x_2 - x_1}$ Used to calculate the rate of change (slope) between two points $(x_1, y_1)$ and $(x_2, y_2)$. If the slope calculated between any two pairs of points in a function's table is constant, the function is linear. Characteristics of Nonlinear Equations Equations that are not in the form $y = mx + b$. This often includes variables with exponents other than 1 (e.g., $x^2$, $x^3$), variables in the den...