Functions - Teacher Copy - Greatest Integer

8/3/2019 Functions - Teacher Copy - Greatest Integer

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Step FunctionsOctober 19, 2011


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The Concept

A step functionis a function that is constant atintervals, but changes abruptly for certain values of

the independent variable, called critical values.

The graph is made up of horizontal segments, usually

closed at one end and open at the other. Since the graph cannot be drawn without lifting the

pencil, the function is called discontinuous.


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Real Life Example

1) Federal income tax:

Taxable annual

income ($)

Tax Rate

(%)

029 590 17

29 59059 180 26

59 180 - 29

What do the open dots mean?

Greater than

What do the closed dots mean?

Less than or equal to


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Real life Examples

Parking Garage

Cell Phone Plans

Taxi fare

Postage

Sales


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Basic Step Function

= Also known as theGreatest Integer Function


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Basic Step Function

Its graph contains half-open segments

that are 1 unit long and 1 unit apart.


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[ what do these mean ] ?

The GREATEST INTEGER brackets tell you to

ALWAYS round down to the integer.

Ask yourself what is the biggest integer in the square

brackets?

For example:

1.99 = 1 = 3


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Transformed

= +

Turn to page 125 in ReflectionsTake a few minutes to check

your parameters.


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Parameter

Affects the distance between the linesegments

If > 1 the distance is greaterIf0 < < 1 the distance is smaller

Affects the slope of the step

Ifa is negativereflects about the -axis


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Parameter

Affects the length of the segments

If > 1 the segments are shorterIf0 < < 1 the segments are longer

Affects the slope of the step

Ifb is negativereflects about the -axis


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Parameter and

Still the vertex of the line segments,where is the first solid dot?

Consider this the first solid dot usuallyoriginating on (0,0)


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To Graph

1. Find:

a) The distance between the segments ;

b) The length of the segments

and their endpoints

(open or closed depending on the sign of b);

c) The horizontal () and the vertical () translations.

2. The images of the points (0, 0), (1,0) and (1,1) canalso be found by performing calculations on the

coordinates and generalizing the graph.


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Using TI-83

Access int( usingMATH - NUM 5 Enter the function.

Without DOT

(not good). MODE

Change

CONNECTED

to

DOT


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Graph = 2

3 + 4

Which geometric

transformations are

associated with thisgraph?


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Properties

Domain:

Range: {,-2, 0, 2, 4, 6, }

Variation: decreasing over

Sign: positive over ,5 negative over 7,

Zeros: 5, 7

Initial Value: 6


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Finding the Rule

To find the rule, you need to be VERYfamiliar with your

parameters.

You look at the ways the graph has been transformed, andapply those to the graph.


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An example

A stores accountant calculates the weekly commission

paid to its sales staff using the rule = 30

,

where is the weekly sales total of the salesperson and(), the size of the commission. Both of these amountsare in dollars.

1. What is the minimum weekly sales total a sales person

must attain to earn a commission?


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Store accountant

What is the minimum weekly sales total a sales personmust attain to earn a commission?

= 30

1000

Logic:

Knowledge of Greatest Integer tells us that if we wantcommission (()) to be more than zero, the

1

For this to happen = 1000


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Store accountant



,


2. How much commission will a saleswoman earn if her

weekly sales total is $12 354.45?


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Store accountant

2. How much commission will a saleswoman earn if her weeklysales total is $12 354.45?

= 30

1000

= 30

12354.45

1000

= 30 12.35445

= 30 12 = $360


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Store accountant



,


3. How much must a salesman sell in a week to earn a

commission of at least $120?


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Store accountant

3. How much must a salesman sell in a week to earn a commission ofat least $120?

= 30

1000

120 30

1000

4 1000

4

1000

4000 x


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Store accountant



,


4. Under what circumstances will a sales person earn a

commission of exactly $100?


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Store accountant

4. Under what circumstances will a sales person earn a commission ofexactly $100?

= 30

1000

100 = 30

1000

3.33 =

1000

A Greatest Integer cannot be a decimal value therefore NEVER!


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Practice - Reflections

Page 126# 1, 2, 3, 4, 6, 8, 9, 12, 14, 15, 23

Documents

Functions - Teacher Copy - Greatest Integer