4 PIECEWISE FUNCTIONS What You Should Learn I

#4 PIECEWISE FUNCTIONS

What You Should Learn: ① I can graph any piecewise function. ① I can evaluate piecewise functions from multiple representations. ① I can distinguish between inclusive and exclusive points. ① I can write a piecewise function from multiple representations.

Graphing & Evaluating Piecewise Functions – Day 1 Today’s objective: ①I can graph any piecewise function. ①I can evaluate piecewise functions from multiple representations. ①I can distinguish between inclusive and exclusive points.

Definition of Piecewise • Root word is Piece.

Piecewise Functions: Graphically • Piecewise Functions are Pieces of different functions graphed together on the same coordinate plane. • Each of these Pieces is defined on a different domain. (x intervals) • Graphs are not continuous.

Examples of Piecewise Functions: Graphically

Piecewise Functions: Algebraically • When functions are defined by more than one equation, they are called piece-wise defined functions. • Each equation is a different Piece.

Examples of Piecewise Functions: Graphically

For the following function a) Find f(-1), f(3). b) Find the domain. c) Sketch the graph.

For the following function a) f(-1) = -1 + 3 = 2 f(1) = 3 f(3) = -3 + 3 = 0.

b) Find the domain. -2 -1 0 1 Domain of f(x): [-2, ∞)

c) Graph

c) Graph

Creating Piecewise Equations – Day 2 Today’s objective: I can write a piecewise function from multiple representations. (From a Graph)

Example 1: Write the function for each graph.

Example 2: Write the function for each graph.

Example 3: Write the function for each graph.

Example 5: Write the function for each graph.

Creating Piecewise Functions from Real World Scenarios – Day 3 Today’s objective: I can write a piecewise function from multiple representations. (From a Real World Scenario)

4 -Step Problem Solving Process: STEP 1: Understand the problem: a) Read the entire problem. b) Can you restate the problem?

4 -Step Problem Solving Process: STEP 2: Devise a Plan: a) Highlight any given information. b) Eliminate any unnecessary info. c) Define the variable using the unknown info. d) Relate the given info to the unknown info with a formula or equation.

4 -Step Problem Solving Process: STEP 3: Execute the Plan: a) Model the problem with the equation. b) Solve the equation for the unknown. c) Be sure to label the units.

4 -Step Problem Solving Process: STEP 4: Check Your Work: a) Check the solution in the original equation. b) Does your answer make sense in the context of the problem?

Example 1: Create a Piecewise Function You go to Wal-Mart to buy some candy. You decide to buy snickers because they have a special deal on snickers. A bag of snickers costs $3. 45, but if you buy 4 or more bags, they only cost $3. 00 per bag. Create a piecewise function to represent the cost of the bags of snickers.

Solution: Example 1 • b = number of bags of snickers • C(b) = Cost of all bags of snickers purchased • $3. 45 = cost of a bag if less than 4 bags are purchased • $3. 00 = cost of a bag if 4 or more bags are purchased.

Example 2: Create a Piecewise Function The Mad Hatter is ordering cups from Teacups, Limited, for his tea party. The Teacups, Limited catalog prices cups according to the number of cups ordered. For orders of 20 or fewer cups, the price is $1. 40 per cup plus $12 shipping and handling on the order. For orders of more than 20 cups, the price is $1. 10 per cup plus $15 shipping and handling. a) Create a function to describe the price of cups.

Example 2: Create a Piecewise Function b) How much will it cost the Mad Hatter to order 16 cups? c) If the Mad Hatter wants to spend at most $45, what is the maximum number of cups he can order?

Solution: Example 2 • c = the number of cups ordered • P(c) = the price of all cups ordered • $1. 40 cost of each cup if 20 or less are ordered • $12. 00 shipping if 20 or less are ordered • $1. 10 cost of each cup if more than 20 are ordered • $15. 00 shipping if more than 20 are ordered b) $34. 40 c) 27 cups

Example 3: Create a Piecewise Function Every month your cell phone plan costs $75 and gives you unlimited talk, 500 text messages, and no data plan. It costs $0. 10 per text message sent in excess of the 500 that you are originally allotted. a) Write a piecewise function to determine the amount of your cell phone bill. b) How much will it cost you if you send 750 text messages?

Solution: Example 3 • $75 = monthly cost of cell phone bill • $0. 10 = cost of each text in excess of the plan’s allotted 500. • t = number of text messages sent • T-500 = number of texts in excess of 500 allotted • C(t) = Cost of your cell phone bill b) $100

Example 4: Create a Piecewise Function A construction worker earns $17 per hour for the first 40 hours of work and $25. 50 per hour for work in excess of 40 hours. a) Create a function to represent the amount of her paycheck. b) One week she earned $896. 75. How much overtime did she work?

Solution: Example 4 • $17 = hourly pay: hours up to 40. • $25. 50 = hourly pay; hours in excess of 40 • h = # of hours worked • h-40 =# hours worked in excess of 40 • P(h) = total amount of paycheck b) 8. 5 overtime hours

Example 5: Create a Piecewise Function Southeast Electric charges $0. 09 per kilowatt-hour for the first 200 k. Wh. The company charges $0. 11 per kilowatt-hour for all electrical usage in excess of 200 k. Wh. a) Create a function to model this scenario. b) How many kilowatt-hours were used if a monthly electric bill was $57. 06?

Solution: Example 5 • h = # kilowatt-hours • C(h) = Cost of total kilowatt-hours b) 555. 09 Kwh

YOU TRY: On Your Whiteboard! Ex. 1: A city parking lot uses the following rules to calculate parking fees: ① A flat rate of $5. 00 for any amount of time up to and including the first hour. ① A flat rate of $12. 50 for any amount of time over 1 hour and up to and including 2 hours. ① A flat rate of $13 plus $3 per hour for each hour after 2 hours. Create a piecewise function to model the parking fees.

YOU TRY: On Your Whiteboard! Ex. 2: Your job pays you $8. 50 an hour for a normal 40 hour work week. If you work over 40 hours then you get paid overtime, at time and half (1. 5 times your normal rate) for anything over 40 hours, but you are not allowed to work more than 20 overtime hours. a) Create a function that models this scenario. b) How much would your paycheck be if you worked 45 hours?

YOU TRY: On Your Whiteboard! Ex. 3: Every month your cell phone plan costs $230 and gives you unlimited talk, text messages, and 2 GB of data. Extra data for each month is $15 per GB you go over your allotted 2 GB. a)Write a piecewise function to represent this situation.

YOU TRY: On Your Whiteboard! Ex. 4: You are teaching tomorrow with an activity and you want to use candy to motivate your students. You go to Fred’s Food Club and they have a special going on for blow pops. For 2 or less bags it cost $4. 35 for each bag, but if you buy more than 2 bags you get the special price of $3. 25 per bag. a) Write a piecewise function to represent this situation.

YOU TRY: On Your Whiteboard! Ex. 5: Greenville Utilities charges a basic customer charge of $10. 99 and $0. 1260 per k. Wh for 400 k. Wh or less. If you go over 400 k. Wh you will then pay $0. 1151 per k. Wh for all k. Wh in excess of the original 400 KWh. a) Write a piecewise function to represent this situation.