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Piecewise Functions and Evaluating Them

This lesson provides an introduction to piecewise functions and how to evaluate them. It also explores real-life applications of piecewise functions and teaches how to determine open and closed points on the graph.

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Piecewise Functions and Evaluating Them

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  1. Piecewise Functions 6-3 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

  2. 1 2 1 y = – x + 43 2 Warm Up Write the equation of each line in slope-intercept form. 1. slope of 3 and passes through the point (5, 30) y = 3x + 2 2. slope of – and passes through the point (6, 40)

  3. Warm Up For each function, determine whether the graph opens upward or downward. downward 1. f(x) = -4x2 + 6x + 1 upward 2. f(x) = 8x2 – x - 2 Write each function in slope-intercept form. y = -3x + 10 3. Y + 3x =10 4. -6y – 12x = 24 y = -2x - 4

  4. Objectives Evaluate piecewise functions. Use piecewise functions to describe real-world situations.

  5. How can you tell where to put open and closed points on the graph of a piecewise function? Essential Question

  6. 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).

  7. Remember! When using interval notation, square brackets [ ] indicate an included endpoint, and parentheses ( ) indicate an excluded endpoint. (Lesson A-1)

  8. Example 1: Consumer Application 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.

  9. Example 1 Continued The domain of the function is divided into three intervals: [0, 2) Weights under 2 Weights 2 and under 5 [2, 5) [5, +∞) Weights 5 and over

  10. Example 1 Continued Step 2 Write a verbal description. Mixed nuts cost $8.00 per pound for less than 2 lb, $6.00 per pound for 2 lb or more and less than 5 lb, and $5.00 per pound for 5 or more pounds.

  11. Check It Out! Example 1 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.

  12. Check It Out! Example 1 Continued The domain of the function is divided into three intervals: [8, 12) $28 [12, 4) $24 [4, 9) $12

  13. Check It Out! Example 1 Continued 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.

  14. 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. The rule for the movie ticket prices from Example 1 on page 422 is shown.

  15. 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.”

  16. 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.

  17. Example 2A: Evaluating a Piecewise Function Evaluate each piecewise function for x = –1 and x = 4. 2x + 1 if x ≤ 2 h(x) = x2 – 4 if x > 2 Because –1 ≤ 2, use the rule for x ≤ 2. h(–1) = 2(–1) + 1 = –1 Because 4 > 2, use the rule for x > 2. h(4) = 42 – 4 = 12

  18. g(–1) = 2(–1) = 1 2 Example 2B: Evaluating a Piecewise Function Evaluate for x = -1 and x = 4 2x if x ≤ –1 g(x) = 5x if x > –1 Because –1 ≤ –1, use the rule for x ≤ –1. Because 4 > –1, use the rule for x > –1. g(4) = 5(4) = 20

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

  20. Check It Out! Example 2b Evaluate each piecewise function for x = –1 and x = 3. 3x2 + 1 if x < 0 g(x) = 5x –2 if x ≥ 0 Because –1 < 0, use the rule for x < 0. g(–1) = 3(–1)2 + 1 = 4 Because 3 ≥ 0, use the rule for x ≥ 0. g(3) = 5(3) – 2 = 13

  21. How can you tell where to put open and closed points on the graph of a piecewise function? An open endpoint indicates a value not included in the domain and a closed endpoint indicates a value that is included in the domain. Essential Question

  22. Teacher: This piecewise function begins with high profit but shows big losses as it continues. Student: Sound like a piece-unwise function to me.

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