Do Now 10/23/2015 Write the equation of each line in slope- intercept form. 1. slope of 3 and passes through the point (50, 200) y = 3x slope of – and passes through the point (6, 40) 1 2 y = – x

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Write and graph piecewise functions. Use piecewise functions to describe real-world situations. Objectives

piecewise function step function Vocabulary

A piecewise function is a function that is a combination of one or more functions. The rule for a piecewise function is different for different parts, or pieces, of the domain. For instance, movie ticket prices are often different for different age groups. So the function for movie ticket prices would assign a different value (ticket price) for each domain interval (age group).

When using interval notation, square brackets [ ] indicate an included endpoint, and parentheses ( ) indicate an excluded endpoint. (CC1) Remember!

Create a table and a verbal description to represent the graph. Step 1 Create a table Because the endpoints of each segment of the graph identify the intervals of the domain, use the endpoints and points close to them as the domain values in the table. Check It Out! Example 1

The domain of the function is divided into three intervals: $28 [8, 12) [12, 4) [4, 9) $24 $12 Check It Out! Example 1 Continued Green Fee ($) Time Range (h) 288 A.M. – noon 24 noon – 4 P.M P.M. – 9 P.M.

Step 2 Write a verbal description. The green fee is $28 from 8 A.M. up to noon, $24 from noon up to 4 P.M., and $12 from 4 up to 9 P.M. Check It Out! Example 1 Continued

12 if x < –3 20 if x ≥ 6 f(x) = Because –3 ≤ –1 < 6, use the rule for –3 ≤ x < 6. f(–1) = 15 Check It Out! Example 2a Evaluate each piecewise function for x = –1 and x = if –3 ≤ x < 6 f(3) = 15 Because –3 ≤ 3 < 6, use the rule for –3 ≤ x < 6.

3x if x < 0 5x – 2 if x ≥ 0 g(x) = Because –1 < 0, use the rule for x < 0. Because 3 ≥ 0, use the rule for x ≥ 0. g(3) = 5(3) – 2 = 13 g(–1) = 3(–1) = 4 Check It Out! Example 2b Evaluate each piecewise function for x = –1 and x = 3.

You can graph a piecewise function by graphing each piece of the function.

f(x) = Graph the function. 4 if x ≤ –1 –2 if x > –1 The function is composed of two constant pieces that will be represented by two rays. Because the domain is divided by x = –1, evaluate both branches of the function at x = –1. Check It Out! Example 3a

O ● Check It Out! Example 3a Continued The function is 4 when x ≤ –1, so plot the point (–1, 4) with a closed circle and draw a horizontal ray to the left. The function is –2 when x > –1, so plot the point (–1, –2) with an open circle and draw a horizontal ray to the right.

g(x) = Graph the function. x + 3 if x ≥ 2 –3x if x < 2 Do Now 10/26/2015

g(x) = Graph the function. x + 3 if x ≥ 2 –3x if x < 2 The function is composed of two linear pieces. The domain is divided at x = 2.

Add an open circle at (2, –6) and a closed circle at (2, 5) and so that the graph clearly shows the function value when x = 2. x–3xx + 3 –412 – –65 47 O ●

g(x) = Graph the function. x + 2 if x ≥ 3 –2x if x < 3 Do Now 10/26/2015

Piecewise functions are not necessarily continuous, meaning that the graph of the function may have breaks or gaps. To write the rule for a piecewise function, 1.determine where the domain is divided 2.Write a separate rule for each piece 3.Combine the pieces by using the correct notation.

Check It Out! Example 3 Shelly earns $8 an hour. She earns $12 an hour for each hour over 40 that she works. Sketch a graph of Shelly’s earnings versus the number of hours that she works up to 60 hours. Then write a piecewise function for the graph. Step 1 Make a table to organize the data. Shelly’s Earnings Hours worked Pay ($/hr) 0–408 >4012 R U HAPPY now?

Step 2 Because the number of hours worked is the independent variable, determine the intervals for the function. 0 ≤ h ≤ 40 h > 40 She works less than or equal to 40 hours. Check It Out! Example 3 Continued She works more than 40 hours. What is an independent variable?

Step 3 Graph the function. Shelly earns $8 per hour for 0–40. After 40 hours, she earns $12 per hour. Check It Out! Example 3 Continued

Step 4 Write a linear function for each leg. Use a slope-intercept form: y = mx + b 0–40 hours: $8 per 1 h Hours > 40: $320 & $12 per h Use m = 8. Use m = 12 and b = 320. The function rule is f(h) = 8h if 0 ≤ h ≤ 40 12h if h > 40 Check It Out! Example 3 Continued

Practice/ HW Worksheet: Practice B – Piecewise Function Due by 10/27