Linear Functions and Their Components

A linear function exhibits a constant rate of change, expressed by the standard equation $y = mx + b$. Here, $y$ is the dependent variable, $x$ is the independent variable, $m$ represents the slope (rate of change), and $b$ is the y-intercept.

Key Components

• Slope ($m$)

  • Defines the rate of change: how much $y$ changes per unit change in $x$.

  • Calculated as: $m = rac{ ext{rise}}{ ext{run}}$ (vertical change over horizontal change).

• Y-Intercept ($b$)

  • The point where the line intersects the y-axis.

  • Occurs when $x=0$, simplifying the equation to $y = b$.

Graphing Linear Functions

• To graph, select at least two $x$ values, calculate their corresponding $y$ values, plot these points, and draw a straight line through them, extending in both directions. Label axes and the y-intercept ($b$) for clarity.