Greatest Integer Function.notebook
January 11, 2017
Greatest Integer Function
Greatest Integer
Notation:
(sometimes
).
To evaluate
, drop the brackets
the real
number
than or equal to
Examples
:
(GIF)
,
,
and replace
with the greatest integer
less
the number.
= 9
= 8
= 6
= 0
= 4
= 1
1
Greatest Integer Function.notebook
January 11, 2017
Basic GIF
Dom:
Ran:
Rule:
Graph:
y
Inc:
Dec:
5
4
Never
3
2
1
x
­5
­4
­3
­2
­1
0
­1
­2
1
2
3
4
Pos:
Neg:
5
Max: None
Min: None
­3
­4
­5
y­intercept: 0
x­intercept: 2
Greatest Integer Function.notebook
January 11, 2017
Recall that in a basic function...
Parameters a and b are both equal to 1
and
Parameters h and k are both equal to 0.
3
Greatest Integer Function.notebook
January 11, 2017
Transformed GIF
Rule:
Parameter a
y
y
6
6
5
5
4
4
3
3
2
2
1
­6
­5
­4
­3
­2
­1
0
­1
1
x
1
2
3
4
5
6
­6
­5
­4
­3
­2
­1
x
0
1
2
3
4
5
6
­1
­2
­2
­3
­3
­4
­4
­5
­5
­6
­6
4
Greatest Integer Function.notebook
January 11, 2017
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
x
0
­1
­2
­3
­4
­5
­6
1
2
3
4
5
6
Summary: Parameter a determines the distance between each step (called the counterstep).
The counterstep = If a + , the function is increasing.
If a ­ , the function is reflected about the x­axis and is decreasing.
Steps:
5
Greatest Integer Function.notebook
January 11, 2017
Parameter b
y
y
6
6
5
5
4
4
3
3
2
2
1
­6
­5
­4
­3
­2
­1
1
x
0
1
2
3
4
5
6
­6
­5
­4
­3
­2
­1
0
­1
­1
­2
­2
­3
­3
­4
­4
­5
­5
­6
­6
length
x
1
2
3
4
5
6
length
6
Greatest Integer Function.notebook
January 11, 2017
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
x
0
­1
­2
­3
­4
­5
­6
1
2
3
4
5
6
Summary: Parameter b determines the length of each step.
The length = i.e., the
reciprocal of
b.
If b + , the function is increasing.
Steps: If b ­ , the function is decreasing and is reflected about the y­axis.
Steps: 7
Greatest Integer Function.notebook
January 11, 2017
If a and b are both negative, the function is increasing again, but each step starts with an empty dot and ends with a full dot.
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
x
0
1
2
3
4
5
6
­1
­2
­3
­4
­5
­6
8
Greatest Integer Function.notebook
Remember:
January 11, 2017
If a and b have the same sign , or if
then the function is increasing :
If a and b have opposite signs , or if
,
,
then the function is decreasing :
9
Greatest Integer Function.notebook
Notice that the origin,
January 11, 2017
, has been a solid (coloured)
dot.
Parameter h translates the function h units left or right;
Parameter k translates the function k units up or down.
This means that
will always be a solid dot .
10
Greatest Integer Function.notebook
January 11, 2017
Example:
y
9
8
7
6
5
4
3
2
1
}
­6
­5
­4
­3
­2
­1 ­10
­2
­3
­4
­5
­6
­7
­8
­9
x
1
2
3
4
5
6
11
Greatest Integer Function.notebook
January 11, 2017
Using all of the information we know about a, b, h & k,
we
will graph Greatest Integer Functions and find the
equation of a GIF from a given graph.
12
Greatest Integer Function.notebook
Examples: 1)
January 11, 2017
Graph
y
1
x
1
2
13