1. Warm Up What is an integer? Determine the largest integer that “fits” into the following values: 4.556.017.99 Challenge: - 3.5

2. 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 46.01 67.99 7 Challenge: - 3.5 -4

4. Greatest Integer Functions Greatest integer functions use the following notation: This is the greatest integer machine! Put a number in, get an integer out! any number integer

5. Use the following equations to make a statement about what the greatest integer machine does to a number. 3.15 3 18.99 18 -8.5 -9 -17.2 -18 -6.99 -7 101.25 101

6. 012345678-2-3-4-5-6-7 When you use the greatest integer function, the answer is the integer on the immediate left on the number line. 6.31-6.31 Exception: When you evaluate an exact integer, like 3, the answer is the integer itself. 3

7. Example 1 2 = 9 = -3 = 2 ? 9 ? ? 2.2 = 1/21/2 = -4.1 = 2? 0 ? -5 ?

8. 9.1 = 51/351/3 = -2 2 / 9 = 9 ? 5 ? -3 ? Example 2

9. If there is an operation inside the greatest integer brackets, it must be performed before applying the function.

10. Example 3

11. xy 0.25.5.75 1 1.25 1.5 1.75 2 Graphing Greatest Integer Functions

12. xy 0.25.5.75 1 1.25 1.5 1.75 2 What if we kept going?

13. 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.

14. Rather than place a long series of points on the graph, a line segment can be drawn for each step.

15. The graphs shown thus far have been magnified to make a point. However, these graphs are usually shown at a normal scale.

16. How would we shift the following graph? Example 4: Translating Greatest Integer Functions

18. Example 5

19. 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.