Linear Functions

There are two main forms for linear functions you should be aware of for your calculus course.

1. Slope-Intercept Form: $y=mx+b$
   • $m$ is the slope
   • $b$ is the $y$-intercept
2. Point-Slope Form: $y-y_{0}=m(x-x_{0})$
   • $m$ is the slope
   • $(x_{0},y_{0})$ is any point on the line

In addition to this, there are two special cases you should know.

1. Horizontal Line: $y=b$
   • $m=0$ is the slope
   • $b$ is the $y$-intercept
2. Vertical Line: $x=a$
   • the slope $m$ is undefined
   • $a$ is the $x$-intercept

Slope

Given two points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ on a line, you can calculate the slope of the line using $$ m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}. $$ We can classify lines based on their slopes. Let $m_{1}\neq0$ and $m_{2}\neq0$ be the slopes of two lines.

1. The two lines are parallel if their slopes are equal, or $m_{1}=m_{2}$.
2. The two lines are perpendicular if their slopes are negative reciprocals, or $m_{1}= \displaystyle -\frac{1}{m_{2}}$.