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

3. Homework Check

4. Unit 1, Day 6

5. 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

6. 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

7. 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

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

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

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

11. Example 3

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

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

14. When all these points are strung together the graph looks something 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.

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

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

17. Just remember: Closed point on the left, open point on the right! Graphing Greatest Integer Functions in the Calculator You can use the “int( )” function in your calculator to graph greatest integer functions.

18. Example 4 Use your calculator to graph the following function:

19. Example 5: Wheels Bike Rentals charges a $6.00 flat rate and $1.50 for each hour you rent a bicycle not including fractions of an hour. Use the greatest integer function to create a model for the cost C of renting a bicycle for x hours. Sketch the graph for up to 5 hours. xC