![Page 1: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/1.jpg)
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
![Page 2: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/2.jpg)
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
![Page 3: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/3.jpg)
![Page 4: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/4.jpg)
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
![Page 5: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/5.jpg)
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
![Page 6: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/6.jpg)
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
![Page 7: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/7.jpg)
Example 1
2 =
9 =
-3
=
2?9?
-3?
2.2 =
1/2 =
-4.1=
2?0?-5?
![Page 8: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/8.jpg)
9.1 =
51/3 =
-22/9 =
9?5?
-3?
Example 2
![Page 9: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/9.jpg)
If there is an operation inside the greatest integer brackets, it must be performed before applying the function.
xxf 5.5)(
)3(f
![Page 10: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/10.jpg)
)6.3(.2)4.1(.1
ff
Example 3
xxf 5.5)(
![Page 11: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/11.jpg)
x y0
.25.5
.751
1.251.5
1.752
Graphing Greatest Integer Functions
xy
![Page 12: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/12.jpg)
x y0
.25.5
.751
1.251.5
1.752
What if we kept going?
![Page 13: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/13.jpg)
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.
![Page 14: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/14.jpg)
Rather than place a long series of points on the graph, a line segment can be drawn for each step.
![Page 15: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/15.jpg)
The graphs shown thus far have been magnified to make a point. However, these graphs are usually shown at a normal scale.
![Page 16: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/16.jpg)
How would we shift the following graph?
342 xy
Example 4: Translating Greatest Integer Functions
![Page 17: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/17.jpg)
32 xy
![Page 18: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/18.jpg)
221
xyExample 5
![Page 19: Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5](https://reader036.vdocuments.net/reader036/viewer/2022062401/5a4d1b577f8b9ab0599a9d0e/html5/thumbnails/19.jpg)
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.