Greatest Integer Function (Step Function)

The greatest integer of a real number x is represented by [x] and Definition The greatest integer of a real number x is represented by [x] and {

Examples [4.7] = 4 [17] = 17 [-5.1] = -6 [0.9] = Round down to nearest integer [-5.1] = -6 [0.9] =

Basic Greatest Integer Function

Example The cost of parking is free for less than an hour and $1 per hour thereafter. f(x)=[x]

Standard Equation Where each step in the graph has a length of and a height of

Parameters A & B Parameter A: Changes steps in a vertical stretch or contraction Parameter B: Changes steps in a horizontal stretch or contraction

Parameters A & B If b > 0, the steps are closed on the left and open on the right If b < 0, the steps are open on the left and closed on the right

Parameters A & B If ab > 0, the function is increasing If ab < 0, the function is decreasing

Four Cases ④ a < 0, b < 0 ① a > 0, b > 0

f(x) has zeros if and only if k is a multiple of a Ex.

Zeros Since Then solve

Zeros

Domain: Range: Initial value: Sign: Other Properties Domain: Range: Initial value: Sign: Graph the function and determine when

Graph and find all properties of Example Graph and find all properties of

Example Step Length: Step height:

Example Zeros:

Example Init Value: Domain: Range:

Example

Homework Workbook P. 123 #1 P. 124 #4 P. 126 #10, 11 &12