Chapter 2: Functions and Models Lesson 7: Step Functions Mrs. Parziale

Vocabulary The greatest integer function is the function f such that for every real number x, f(x) is the greatest integer less than or equal to x. It is written as Also known as Floor Function or Rounding Down function

Example 1: You can fit 12 cans of soda in a box. Make a graph of cans (x) vs. number of complete boxes (y). Is this a function? Why / why not?

Greatest Integer Function It is written Floor Function Rounding Down function Also known as the Floor Function or Rounding Down function. Round down left Round down to the integer to the left on the number line. piecewise linear The graph of this is piecewise linear. Each step is part of a horizontal line.

Example 2: Evaluate: a) = ______ b) = ______ c) = ______

Evaluate: Using your Calculator’s MATH function. (Math Num Int ( ) a)INT (3.7) = ______ b) INT (  ) = ______ c) INT (-  ) = ______ Example 2, cont.:

Example 3: Graph X What are the domain and range? Start by examining the graph and then making a table.

discontinuous This graph is discontinuous. points of discontinuity The values of x where you lift your pencil off the paper are the points of discontinuity. X

ON A CALCULATOR (TI83 or TI83+): We use the INT function (Math Num int( ) y = int(x) It looks connected, but it is not. TI84 does not show this “connection.”

Rounding Up or Ceiling function: is the smallest integer greater than or equal to x. Note:

Example 2: Evaluate: a) = ______ b) = ______

Example 5: How many buses (b) are needed to transport (s) students if each bus can hold 44 students?

Example 6: Suppose it costs $50 to rent a bus in example 5. What will it cost to transport 300 students?

Example 7: Graph

Closure How are the greatest integer and the ceiling function different? What does the open circle and closed circle mean in either function? How would you find the and ? Is the graph of the greatest integer a function?