warm up what is an integer? determine the largest integer that “fits” into the following values:...
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Warm Up
What is an integer?
Determine the largest integer that “fits” into the following values:
4.55 6.01 7.99
Challenge: - 3.5
Answers
What is an integer?The set of positive and negative natural numbers.
Determine the largest integer that “fits” into the following values:
4.55 4 6.01 6 7.99 7
Challenge: - 3.5 -4
Greatest Integer FunctionsGreatest integer functions use the following
notation:
xy
This is the greatest integer machine! Put a number in, get an integer out!
any number
integer
Use the following equations to make a statement about what the greatest integer machine does to a number.
xy
xy
xy
3.15 3
18.99 18
-8.5 -9
xy
xy
xy
-17.2 -18
-6.99 -7
101.25 101
0 1 2 3 4 5 6 7 8-1-2-3-4-5-6-7
When you use the greatest integer function, the answer is the integer on the immediate left on the number line.
6.31-6.31 32
514
31.6 514
32 3
Exception: When you evaluate an exact integer, like 3, the answer is the integer itself.
3
Example 1
2 =
9 =
-3
=
2?9?
-3?
2.2 =
1/2 =
-4.1=
2?0?-5?
9.1 =
51/3 =
-22/9 =
9?5?
-3?
Example 2
If there is an operation inside the greatest integer brackets, it must be performed before applying the function.
xxf 5.5)(
)3(f
)6.3(.2)4.1(.1
ff
Example 3
xxf 5.5)(
x y0
.25.5
.751
1.251.5
1.752
Graphing Greatest Integer Functions
xy
x y0
.25.5
.751
1.251.5
1.752
What if we kept going?
When all these points are strung together the graph looks like a series of steps.
Notice that the left of each step begins with a closed point, but the right of each step ends with an open point.
Rather than place a long series of points on the graph, a line segment can be drawn for each step.
The graphs shown thus far have been magnified to make a point. However, these graphs are usually shown at a normal scale.
How would we shift the following graph?
342 xy
Example 4: Translating Greatest Integer Functions
32 xy
221
xyExample 5
Example 6: Wheels Bike Rentals charges a $6.00 flat rate and $1.50 for each hour you rent a bicycle. If you ride for part of an hour, you will be charged for an entire hour. Write a greatest integer function to model the situation.