Download - 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
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.
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
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.
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
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.
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