Linear Functions

 A linear function is a function that creates a straight line when graphed. It has the general form:

$$f(x) = mx + b$$

Where:

• $f(x)$ is the dependent variable (often written as $y$),
• $m$ is the slope of the line, which indicates how steep the line is. It is the ratio of the vertical change to the horizontal change between any two points on the line.
• $b$ is the y-intercept, which is the point where the line crosses the y-axis (when $x = 0$).

Key Characteristics of Linear Functions:

1. Constant Rate of Change: The slope $m$ represents the constant rate of change. For every unit increase in $x$, $y$ changes by $m$.
2. Graph: The graph of a linear function is always a straight line.
3. Intercept: The y-intercept $b$ is the value of $y$ when $x = 0$.
4. Slope: The slope $m$ can be positive, negative, or zero:
   • A positive slope means the line rises as you move from left to right.
   • A negative slope means the line falls as you move from left to right.
   • A slope of zero means the line is horizontal.

Examples:

1. $f(x) = 2x + 3$

   • Slope $m = 2$, which means for every 1 unit increase in $x$, $y$ increases by 2.
   • Y-intercept $b = 3$, meaning the line crosses the y-axis at $(0, 3)$.

2. $f(x) = -x + 5$

   • Slope $m = -1$, so for every 1 unit increase in $x$, $y$ decreases by 1.
   • Y-intercept $b = 5$, meaning the line crosses the y-axis at $(0, 5)$.

Would you like help with solving linear functions or graphing them?