Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

Find the output for each input value. Warm Up Find the output for each input value. Input Rule Output x –3x + 2 y –4 14 2 4 –10

Function Rule A: Square the input. Divide by 2. Subtract 3. Problem of the Day Function Rule A: Square the input. Divide by 2. Subtract 3. Function Rule B: Square the input. Subtract 6. Divide by 2. If the input value for each rule is 222, what is the difference of the two output values? Why? 0; they are equivalent rules.

Learn to find patterns to complete sequences using function tables.

Vocabulary sequence term arithmetic sequence common difference geometric sequence

A sequence is an ordered list of numbers A sequence is an ordered list of numbers. Each number in a sequence is called a term. When the sequence follows a pattern, the terms in the sequence are the output values of a function, and the value of each number depends on the number’s place in the list.

n (position in the sequence)‏ You can use a variable such as n, to represent a number’s position in a sequence. 8 6 4 2 y (value of term)‏ 3 1 n (position in the sequence)‏ In an arithmetic sequence, the terms of the sequence differ by by the same nonzero number. This difference is called the common difference. In a geometric sequence, each term is multiplied by the same amount to get the next term in the sequence.

Additional Example 1A: Identifying Patterns in a Sequence Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. -64 -16 -4 -1 y 5 4 3 2 1 n -256 In the sequence -1, -4, -16, -64, ,…, each number is multiplied by 4. -64 ● 4 = -256. Multiply the fourth number by 4. The sequence is geometric. When n = 5, y = -256.

Additional Example 1B: Identifying Patterns in a Sequence Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. 36 41 46 51 y 5 4 3 2 1 n 31 In the sequence 51, 46, 41, 36, ,…, -5 is added each time. 36 + (-5) = 31. Add -5 to the fourth number. The sequence is arithmetic. When n = 5, y = 31.

Check It Out: Example 1A Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. 24 20 16 12 y 5 4 3 2 1 n 28 In the sequence 12, 16, 20, 24, ,…, 4 is added each time. 24 + 4 = 28. Add 4 to the fourth number. The sequence is arithmetic. When n = 5, y = 28.

-27 -9 -3 -1 y 5 4 3 2 1 n Check It Out: Example 1B Tell whether the sequence of y-values is arithmetic or geometric. Then find y when n = 5. -27 -9 -3 -1 y 5 4 3 2 1 n -81 In the sequence -1, -3, -9, -27, ,…, each number is multiplied by 3. -27 ● 3 = -81. Multiply the fourth number by 3. The sequence is geometric. When n = 5, y = -81.

Additional Example 2A: Identifying Functions in Sequences Write a function that describes the sequence. 3, 6, 9, 12,… Make a function table. 12 4 9 3 6 2 1 y Rule n 1 • 3 2 • 3 Multiply n by 3. 3 • 3 4 • 3 The function y = 3n describes this sequence.

Additional Example 2B: Identifying Functions in Sequences Write a function that describes the sequence. 4, 7, 10, 13,… Make a function table. 13 4 10 3 7 2 1 y Rule n 3(1) + 1 3(2) + 1 Multiply n by 3 and add 1. 3(3) + 1 3(4) + 1 The function y = 3n + 1describes this sequence.

Check It Out: Example 2A Write a function that describes the sequence. 5, 6, 7, 8,… Make a function table. 8 4 7 3 6 2 5 1 y Rule n 1 + 4 2 + 4 Add 4 to n. 3 + 4 4 + 4 The function y = 4 + n describes this sequence.

Check It Out: Example 2B Write a function that describes the sequence. 3, 4, 5, 6,… Make a function table. 6 10 5 9 4 8 3 7 y Rule n 7  4 8  4 Subtract 4 from n. 9  4 10  4 The function y = n – 4 describes this sequence.

Additional Example 3: Using Functions to Extend Sequences Holli keeps a list showing her cumulative earnings for walking her neighbor’s dog. She recorded $1.25 the first time she walked the dog, $2.50 the second time, $3.75 the third time, and $5.00 the fourth time. Write a function that describes the sequence, and then use the function to predict her earnings after 9 walks. Write the number of walks she recorded; 1.25, 2.50, 3.75, 5.00. Make a function table.

Additional Example 3 Continued 5.00 4 3.75 3 2.50 2 1.25 1 y Rule n 1 • 1.25 Multiply n by 1.25. 2 • 1.25 3 • 1.25 4 • 1.25 Write the function. y = 1.25n 9 walks correspond to n = 9. When n = 9, y = 1.25 • 9 = 11.25. Holli would earn $11.25 after 9 walks.

Check It Out: Example 3 Jeff keeps a list showing his cumulative earnings for washing cars. He recorded $2.50 the first time he washed a car, $5 the second time, $7.50 the third time, and $10 the fourth time. Write a function that describes the sequence, and then use the function to predict his earnings after 8 car washes. Write the number of car washed he recorded; 2.50, 5.00, 7.50, 10.00. Make a function table.

Check It Out: Example 3 Continued 10.00 4 7.50 3 5.00 2 2.50 1 y Rule n 1 • 2.50 Multiply n by 1.25. 2 • 2.50 3 • 2.50 4 • 2.50 Write the function. y = 2.50n 8 car washed correspond to n = 8. When n = 8, y = 2.50 • 8 = 20. Jeff would earn $20. after 8 car washes.

Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems

Lesson Quiz: Part I Tell whether each sequence of y-values is arithmetic or geometric. Write a function that describes each sequence, and then find y when n = 5. 1. 6, 12, 18, 24,… 2. –3, –2, –1, 0,… 3. 24, 21, 18, 15,… geometric; y = 6n; 30 arithmetic; y = n – 4; 1 arithmetic; y = 27 – 3n; 12

Lesson Quiz: Part II 4. Arisha used 0.5 cups of nuts in the first batch of cookies that she made, 1 cup in the second, 1.5 cups in the third, and 2 cups in the fourth. Write a function to describe the sequence, and then use the function to predict the amount of nuts in the seventh batch of cookies. y = 0.5n; 3.5 cups.

Lesson Quiz for Student Response Systems 1. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. 4, 8, 12, 16, … A. arithmetic; y = 4n; 20 B. geometric; y = 2n; 10 C. arithmetic; y = 4 + n; 9 D. arithmetic; y = 2n + 2; 12

Lesson Quiz for Student Response Systems 2. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. –5, –4, –3, –2, … A. arithmetic; y = n + 6; 11 B. geometric; y = 2n – 6; 4 C. arithmetic; y = n – 6; –1 D. geometric; y = n; 5

Lesson Quiz for Student Response Systems 3. Tell whether the given sequence of y-values is arithmetic or geometric. Identify a function that describes the sequence, and then find y when n = 5. 16, 12, 8, 4, … A. arithmetic; y = 20 – 2n; 10 B. geometric; y = 4n; 20 C. arithmetic; y = 20 – 4n; 0 D. geometric; y = 30 – 6n; 0

Lesson Quiz for Student Response Systems 4. Rita used 2 tablespoons of butter for the first recipe, 4 tablespoons for the second, 6 tablespoons for the third, and 8 tablespoons for the fourth. Write a function to describe the sequence, and then use the function to predict the amount of butter in the sixth recipe. A. y = 2n; 12 tablespoons B. y = 3n; 18 tablespoons C. y = 4n; 24 tablespoons D. y = 2n + 10; 22 tablespoons