Linear Functions

Modeling Representation

Perhaps the simplest family of functions to become acquainted with is the linear functions. Linear functions include the constant functions (same output for every input), the functions that increase at a constant (positive) rate, and the functions that decrease at a constant (negative) rate.

Indeed, it is this constancy that is the best way to recognize a situation in which linear functions will provide a good model. If we think of constant functions as increasing or decreasing at a constant rate of  0  (i.e., they don't increase or decrease at all), then we can describe all linear functions as follows:

Linear functions model constant rates of change.

Linear functions are often found in situations that involve constant accumulation in an arithmetic sequence.

Review of arithmetic and geometric sequences:

Here are some other situtations that should make you think of linear functions:

• An airplane flys at a constant altitude. (height vs. time)
  
  
• A bathtub fills at a constant rate. (volume vs. time)
  
  
  
• A nice, steady acceleration. (velocity vs. time)
  
  
• The regular progress of the tortoise. (distance vs. time)
  
  
• Weekly deposits of equal paychecks. (amount vs. time)

Words and phrases like "constant," "constant rate," "steady," "regular," and "equal change over equal intervals" are all recognizable features on the faces of the linear family.