Unit 4 Day 5 Piecewise Functions

Warm-up

Warm-up

Homework Answers p. 6

Homework Answers p. 6

Homework Answers p. 7 3) y = x2 4) y = 5) 6) 7)

Homework Packet p. 8, 9, and 10

A function can be in pieces… You can create functions that behave differently depending on the input (x) value.

Piecewise Functions

Example 1: Find: f(-2) = f(3) = -2 + 2 = 0 f(1) = (3 - 1)2 = 4 Top rule -2 + 2 = 0 Bottom rule (3 - 1)2 = 4 Bottom rule (1-1)2 = 0 Domain: all real #s Range: all real #s

Example 2: Find: f(-2) = f(3) = x ≤ -1 so y = -3 f(1) = -(3)2 + 4 = -5 Top rule x ≤ -1 so y = -3 Bottom rule -(3)2 + 4 = -5 Bottom rule -(1)2 + 4 = -3 Domain: all real #s Range: y ≤ 4

Example 3

Example 3: Domain: all real #s Range: all real #s

Example 4: Find: f(-2) = f(3) = x < -1 so y = 3 f(1) = 3 – 4 = -1 Top rule x < -1 so y = 3 Bottom rule 3 – 4 = -1 Top rule (1 + 1)2 - 2 = 2 Domain: all real #s Range: y > -3

Piecewise Functions: Domain and Range (not in notes) Piecewise Functions: Domain and Range

Applications! 1. When a diabetic takes long-acting insulin, the insulin reaches its peak effect on the blood sugar level in about three hours. This effect remains fairly constant for 5 hours, then declines, and is very low until the next injection. In a typical patient, the level of insulin might be modeled by the following function. Here, f (t) represents the blood sugar level at time t hours after the time of the injection. If a patient takes insulin at 6 am, find the blood sugar level at each of the following times. a. 7 am b. 11 am c. 3 pm d. 5 pm 7 – 6 = 1 hour 0 ≤ 1 ≤ 3 40(1) + 100 = 140 11 – 6 = 5 hour2 3 < 5 ≤ 8 220 15:00-6:00 = 9 hours 8 < 9 ≤ 10 -80(9) + 860 = 140 17:00-6:00 = 11 hours 10 < 11 ≤ 24 60

Applications 1 2 3 4 5 5.5 6 6.5 7 4 8 12 16 20 23 26 29 32

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4. A wholesaler charges $3.00 per pound for an order of less than 20 pounds of candy and $2.50 per pound for 20 or more pounds. Write a piecewise function for this situation. Then graph the function. What is the total charge for an order of 15 pounds of candy? For 20 pounds? For 30 pounds? 3(15) = $45.00 2.50(20) = $50.00 2.50(30) = $75.00

Using Technology to graph piecewise Using Technology to graph piecewise. Carefully define each piece in the following way: Enter in Y1: Y1 = (X^2 + 2) ( X ≤ 1) + (-2X + 7) (X > 1) To get the best view of this function, set your window carefully based on your previous sketch or on the table above. You can use the table to check x=1 (where should the open and closed circle be?) Verify that what you graphed by hand is the same as the graph on the calculator screen. Show practice with document camera. Practice: Notes p. 19-21

Homework Packet p. 8, 9, and 10