1. Piecewise Functions

2. Warm Up 1. Can there be two functions on a graph at the same time? 2. If there are, will our graph still be a function? 3. Draw a picture of a graph with more than one function on it. (There isn’t really a right or wrong answer for this. Be creative.) 4. Of all of the things we’ve covered in Unit 2 so far, what types of problems do you want us to definitely go over in review tomorrow?

3. Announcements Today is THE LAST DAY to complete test corrections for the Unit 1 test.

4. Today’s Objectives SWBAT graph piecewise functions. SWBAT evaluate piecewise functions for given domain values. SWBAT identify points of discontinuity and state whether piecewise functions are continuous or not.

5. Piecewise Functions What are they and why do we need them? Functions that have different rules for different domain values. They are made up of 2 or more “normal” functions.

6. Piecewise Functions What are they and why do we need them? They help us represent situations where rates are different at different points in time. (ex) If you are driving a car at 60 mph for the first three hours, then you stop for an hour to eat lunch, then you drive 80 mph for the next four hours. The function would look like this:

7. Can you draw 3 graphs to represent these situations? Saundra is a personal trainer at a local gym. Earlier this year, three of her clients asked her to help them train for an upcoming 5K race. Though Saundra had never trained someone for a race, she developed plans for each of her clients that she believed would help them perform their best. She wanted to see if her plans were effective, so when she attended the race to cheer them on, she collected data at regular intervals along the race. Her plan was to create graphs for each of the runners and compare their performances. Since each had an individualized strategy, each runner ran a different plan during the race. One of her clients (Sue, the oldest one), was supposed to begin slowly, increasing over the first kilometer until she hit a speed which she believed she could maintain over the rest of the race. Her second client, Jim, was supposed to begin with a strong burst for the first kilometer, then slow to a steady pace until the final kilometer when he would finish with a strong burst. Her third client, Jason, is a very experienced runner. His plan was to run at a steady pace for the first two kilometers, then run at his maximum speed for the final 3 kilometers.

8. Sue’s graph One of her clients (Sue, the oldest one), was supposed to begin slowly, increasing over the first kilometer until she hit a speed which she believed she could maintain over the rest of the race.

9. Jim’s graph Her second client, Jim, was supposed to begin with a strong burst for the first kilometer, then slow to a steady pace until the final kilometer when he would finish with a strong burst.

10. Jason’s graph Her third client, Jason, is a very experienced runner. His plan was to run at a steady pace for the first two kilometers, then run at his maximum speed for the final 3 kilometers.

11. Evaluate a piecewise function at specified points of the domain Example:

12. You Try two. f(8) =

13. Continuous or discontinuous? A piecewise function is continuous if:  its functions touch at their meeting points  AND if it is defined for all values of x. To determine whether or not it is continuous:  1. Look to make sure ALL REAL NUMBERS are included in the domain.  2. Evaluate all functions at the end points to see if both functions give the same value.  3. If your piecewise function passes BOTH of the above tests, then it is continuous.

14. Continuous or discontinuous? Example:

15. You try two:

16. Graph a piecewise function

20. Graph a piecewise function. Example

21. You Try the next two.

22. Identify points of discontinuity given a graph. When we are given a graph, we can tell if and where the piecewise function is discontinuous, just by looking at the graph:

23. You try 3.

24. Complete the practice worksheet with a partner. Find 1 partner you would like to work with. Work with your partner to finish the class work sheet. You must work TOGETHER. The first pair with all problems done correctly earns a small prize. Completing this sheet thoroughly earns you your class work stamp for today.

25. Homework Half worksheet – all problems.

26. Presentations Groups that did not complete their presentations on Thursday will complete them today to finish class