1. Piecewise Functions Pre-Cal Advanced Ms. Ouseph

2. Objective for today! We will use our knowledge of domain and transformations to graph piecewise functions.

3. A piecewise function is… A function defined by different equations on different domains.

4. In other words… Think of it as only having a “chunk” of a function!

5. We’ll harp more about that later… For now we will explore an everyday situation where piecewise functions are applicable.

6. arking Garage Suppose we were parking in a parking garage that we had to pay for by the hour. The pay structure is as follows: For the first 4 hours, it costs us $3 per hour to park. For any number of hours parked after 4 hours, it costs $2 per hour to park (These rates are proportional to any amount of time spent parked within the first three hours; for example if we only stayed for ½ hour, we would pay $1.50.) OUR GOAL: Create a function which inputs hours parked and outputs total cost.

7. Total Hours Parked (x)Total Payment (y or f(x)) *For the first 4 hours, it costs us $3/hour to park* 1 2 3 4 Equation 1: *For any number of hours parked after 4 hours, it costs us $2/hour to park* 5 6 7 8 x Equation 2: y = 3x y = 2x + 4 12 + 2(x – 4) f(1) = 3(1) = $3 f(2) = 3(2) = $6 f(3) = 3(3) = $9 f(4) = 3(4) = $12 f(5) = 3(4) + 2(5-4) = $14 f(6) = 3(4) + 2(6-4) = $16 f(7) = 3(4) + 2(7-4) = $18 f(8) = 3(4) + 2(8-4) = $20 In groups, fill out the table below and determine an equation to calculate the total payment with respect to the hourly rate.

8. Equation 1: y = 3x only works when… Equation 2: y = 2x + 4 only works when… Combining these two, we have a piecewise function! The total number of hours parked (x) are between 0-4 0 ≤ x ≤ 4 The total number of hours parked (x) are between greater than 4 x > 4 f(x) = { 3x, 0 ≤ x ≤ 4 2x + 4, x > 4

9. How do we graph a piecewise function?? Using the table we created, we can plot points on a graph to graph the piecewise function! xyxy 0123401234 4567845678 0 3 6 9 12 14 16 18 20 y = 3x, 0 ≤ x ≤ 4 y = 2x + 4, x > 4 Closed Open

10. Each function must live in its own “neighborhood” Let's put up a fence to separate these two “neighborhoods.” y = 3x lives here! y = 2x + 4 lives here! x = 4 Note: Its OK for each neighborhoo d to lie on the fence. They just can’t cross over it!

11. Total Hours Parked Total Payment Parking Hours vs. Payment (0, 0) (1, 3) (2, 6) (3, 9) (4, 12) (5, 14) (6, 16) (7, 18) (8, 20) Both equations are “on the fence,” but one point (from y=3x) is closed while the other (from y=2x+4) is open! f(x) = { 3x, 0 ≤ x ≤ 4 2x + 4, x > 4

12. This analysis is not meant to explain the parking situation mathematically. Rather, it is meant to explain the idea of a piecewise function by showing you where one exists in real life. Now, we will be exploring and graphing other piecewise functions involving an array of functions (linear, quadratic, etc.)!

13. What do we need to know in order to graph these pieces? “Fence,” Domain, Points, Open/Closed, etc…

14. xy xy 1 0 -2 12341234 5 3 1 32103210 Closed Open

15. Now let’s use these table of values to graph our piecewise function! Domain: (-∞, ∞) Range: (-∞, 5] xyxy 1 0 -2 5 3 1 12341234 32103210

16. Let’s look at a quadratic piecewise function! Work with a partner to graph the following function:

17. xyxy 0 -2 -3 01230123 3 2 -6 12341234 ( ) Closed Open ( )

18. Now let’s use these table of values to graph our piecewise function! Domain: (-∞, ∞) Range: (-∞, ∞) xyxy 01230123 12341234 0 -2 -3 3 2 -6

19. In summary… The key to graphing piecewise functions is: – Determining the “fence” This fence will tell you where each piece of the function “lives” – Domain Once you know the domain of each function, its just a matter of creating a table of values to plot points!

20. Now that we have explored various piecewise functions, you will be given time to finish the rest of your worksheet.