Piecewise Functions 2007 MS Mathematics Framework Algebra e

Piecewise Functions

2007 MS Mathematics Framework • Algebra: e. Write, graph, and analyze linear and nonlinear functions (such as quadratic, greatest integer)

Why Piecewise Functions? • What factors are important when determining the price of a phone call? • How do you distinguish the price of a five minute call versus a five and one-half minute call? • How do phone companies use functions to determine your local phone bill?

“Step functions” are a type of piecewise functions. The ceiling function (or least integer function) will round any number up to the nearest integer.

“Step functions” are sometimes used to describe real-life situations. The greatest integer function (or floor function) will round any number down to the nearest integer.

Greatest Integer/Floor Function:

The TI-84 command for the floor function is int (x). Graphing the greatest integer function: The calculator “connects the dots” like a staircase instead of just the steps. The open and closed circles do not show, but you can just see the steps.

Greatest Integer Function • • • [3. 7] = 3 [15. 25] = 15 [4] = 4 [4. 999] = 4 [0. 14] = 0 [-1. 5] = -2

Graphs of the floor & ceiling functions

How Much Is That Phone Call? Phone companies will determine the price of a call by rounding the length of the call to a certain time period (either one minute or six seconds). For instance, a local weekday call on Long D’s Basic Residential plan will cost $0. 25 each minute. Suppose that Long D’s also charges a $0. 15 connection fee for each call. f(x) is the cost of placing a phone call that lasts x minutes Examples minutes cost 65¢ 0<x<1 40¢ 115¢ 1<x<2 65¢ 2<x<3 90¢ 3<x<4 115¢ 4<x<5 140¢ 5<x<6 165¢ 140¢ 215¢

Long D’s Basic Residential Plan

Postage Stamp Function f(x) is the cost of mailing a letter that weighs x ounces Weight cost Examples 0<x<1 32¢ 1<x<2 55¢ 78¢ 2<x<3 78¢ 147¢ 3<x<4 101¢ 4<x<5 124¢ 5<x<6 147¢ 32¢

Postage Stamp Function

How Much Has It Rained?