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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Warm Up Write the equation of each line in slope- intercept form. 1. slope of 3 and passes through the point (50, 200) y = 3x + 50 2. slope of – and passes through the point (6, 40) 1 2 y = – x + 43 1 2

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Write and graph piecewise functions. Use piecewise functions to describe real-world situations. Objectives

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. piecewise function step function Vocabulary

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 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).

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. When using interval notation, square brackets [ ] indicate an included endpoint, and parentheses ( ) indicate an excluded endpoint. Introducing!

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. The domain of the function is divided into three intervals: Weights under 2 [0, 2) [2, 5) [5, ∞) Weights 2 and under 5 Weights 5 and over Example 1

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Create a table and a verbal description to represent the graph. Example 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.

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 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. Example 2

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. The domain of the function is divided into three intervals: $28 [8, 12) [12, 4) [4, 9) $24 $12 Example 2 Continued Green Fee ($) Time Range (h) 288 A.M. – noon 24 noon – 4 P.M. 12 4 P.M. – 9 P.M.

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. A piecewise function that is constant for each interval of its domain, such as the ticket price function, is called a step function. You can describe piecewise functions with a function rule. Read this as “f of x is 5 if x is greater than 0 and less than 13, 9 if x is greater than or equal to 13 and less than 55, and 6.5 if x is greater than or equal to 55.”

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 2x + 1 if x ≤ 2 x 2 – 4 if x > 2 h(x) = Because –1 ≤ 2, use the rule for x ≤ 2. Because 4 > 2, use the rule for x > 2. h(–1) = 2(–1) + 1 = –1 h(4) = 4 2 – 4 = 12 Example 3 Evaluate each piecewise function for x = –1 and x = 4. To evaluate any piecewise function for a specific input, find the interval of the domain that contains that input and then use the rule for that interval.

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. 12 if x < –3 20 if x ≥ 6 f(x) = Because –3 ≤ –1 < 6, use the rule for –3 ≤ x < 6. f(–1) = 15 Example 4 Evaluate each piecewise function for x = –1 and x = 3. 15 if –3 ≤ x < 6 f(3) = 15 Because –3 ≤ 3 < 6, use the rule for –3 ≤ x < 6.

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 5 g(x) = 1 2 Graph each function. x – 3 if 0 ≤ x < 4 x 2 – 3 if x < 0 The function is composed of one linear piece and two quadratic pieces. The domain is divided at x = 0 and at x = 4. (x – 4) 2 – 1 if x ≥ 4

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 5 Continued No circle is required at (0, –3) and (4, –1) because the function is connected at those points.

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. O ● 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. f(x) = Graph the function. 4 if x ≤ –1 –2 if x > –1 Example 6

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Example 6: Sports Application Jennifer is completing a 15.5-mile triathlon. She swims 0.5 mile in 30 minutes, bicycles 12 miles in 1 hour, and runs 3 miles in 30 minutes. Sketch a graph of Jennifer’s distance versus time. Then write a piecewise function for the graph.

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Step 1 Make a table to organize the data. Use the distance formula to find Jennifer’s rate for each leg of the race. Example 6 Continued

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Step 2 Because time is the independent variable, determine the intervals for the function. Swimming: 0 ≤ t ≤ 0.5 Biking: 0.5 < t ≤ 1.5 Running: 1.5 < t ≤ 2 She swims for half an hour. She bikes for the next hour. She runs the final half hour. Example 6 Continued

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Step 3 Graph the function. After 30 minutes, Jennifer has covered 0.5 miles. On the next leg, she reaches a distance of 12 miles after a total of 1.5 hours. Finally she completes the 15.5 miles after 2 hours. Example 6 Continued

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Step 4 Write a linear function for each leg. Use point-slope form: y – y 1 = m(x – x 1 ). Swimming: d = t Biking: d = 12t – 5.5 Running: d = 6t + 3.5 Use m = 0.5 and (0, 0). Use m = 12 and (0.5, 0.5). Use m = 6 and (1.5, 12.5). The function rule is d(t) = t if 0 ≤ t ≤ 0.5 12t – 5.5 if 0.5 < t ≤ 1.5 6t + 3.5 if 1.5 < t ≤ 2 Example 6 Continued

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Independent Practice Due Tomorrow at Beginning of Class p. 426 # 9-18 all

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Lesson Quiz: Part I 1. Graph the function, and evaluate at x = 1 and x = 3. p(x) = x 2 + 2 if x ≤ 2 1 2 x + 3 if x > 2 1 2

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5.2 Piecewise Functions CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. CC.9-12.A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Lesson Quiz: Part II 2. Write and graph a piecewise function for the following situation. A house painter charges $12 per hour for the first 40 hours he works, time and a half for the 10 hours after that, and double time for all hours after that. How much does he earn for a 70-hour week?

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