Absolute value functions are functions that represent the distance between a number and zero on a number line. They can be written as f(x) = |x|, where f(x) is the output (y-value) and x is the input (x-value).
A vertical tangent line is a line that is perpendicular to the curve of a function at a specific point and has an undefined slope. It occurs when there is an abrupt change in direction or when the slope becomes infinite.
In calculus, smooth refers to functions that are continuous and have no sharp corners or breaks in their graph. They exhibit gradual changes and have no abrupt changes in direction.
Piecewise Function: A piecewise function is made up of different equations defined for different intervals or pieces of the domain. It's like having multiple gears on a bike - each gear represents a different equation used to describe the function within its specific interval.