Sequences F.LE.1, 2, 5 F.BF.1, 2 A.SSE.1.a F.IF.1, 2, 3, 4 http://www.cde.ca.gov/be/st/ss/documents/ccssmathstandardaug2013.pdf

Graphs of Sequences: Arithmetic Consider the sequence on page 252: 𝑎 𝑛 =−10+4 𝑛−1 Complete the table and then use the table to graph the ordered pairs on page 253. Complete all parts a-f.

Graphs of Arithmetic Sequences What do you notice about the graph of this arithmetic sequence? What did you notice about all the arithmetic sequences from the graphing activity? It appears that arithmetic sequences form linear functions. Can we prove this to be true? If so, how? Recall: A linear function can be written in slope-intercept form: 𝑓 𝑥 =𝑚𝑥+𝑏 Can we write an arithmetic sequence in this form?

Arithmetic Sequences as Functions Let’s rewrite a generic arithmetic sequence in function form: STATEMENT REASON 𝑎 𝑛 = 𝑎 1 +𝑑 𝑛−1 Explicit Formula for Arithmetic Sequence 𝑓 𝑛 = 𝑎 1 +𝑑 𝑛−1 Represent 𝑎 𝑛 using function notation 𝑓 𝑛 = 𝑎 1 +𝑑𝑛−𝑑 Distributive Property 𝑓 𝑛 =𝑑𝑛+ 𝑎 1 −𝑑 Commutative Property 𝑓 𝑛 =𝑑𝑛+constant Combine Like Terms So 𝑎 𝑛 = 𝑎 1 +𝑑 𝑛−1 can be written as 𝑓 𝑛 =𝑚𝑛+𝑏. All ARITHEMTIC SEQUENCES represent a LINEAR FUNCTION

Graphs of Sequences: Geometric Consider the sequence on page 254: 𝑔 𝑛 = 2 𝑛−1 𝑤ℎ𝑒𝑟𝑒 𝑔 1 =1 Complete the table and then use the table to graph the ordered pairs on page 255. Complete all parts a-f.

Graphs of Geometric Sequences What do you notice about the graph of this geometric sequence? What did you notice about MOST of the geometric sequences from the graphing activity? It appears that most geometric sequences form exponential functions. Can we prove this to be true? If so, how? Recall: An exponential function can be written the form: 𝑓 𝑥 =𝑎∙ 𝑏 𝑥 where a and b are real numbers, and b is greater than 0 but is not equal to 1. Can we write a geometric sequence in this form?

Geometric Sequences as Functions Let’s rewrite a generic geometric sequence in function form: STATEMENT REASON 𝑔 𝑛 = 𝑔 1 ∙ 𝑟 𝑛−1 Explicit Formula for Geometric Sequence 𝑓 𝑛 = 𝑔 1 ∙ 𝑟 𝑛−1 Represent 𝑔 𝑛 using function notation 𝑓 𝑛 = 𝑔 1 ∙ 𝑟 𝑛 ∙ 𝑟 −1 Product Rule of Exponents 𝑓 𝑛 = 𝑔 1 ∙ 𝑟 −1 ∙ 𝑟 𝑛 Commutative Property 𝑓 𝑛 = 𝑔 1 ∙ 1 𝑟 ∙ 𝑟 𝑛 Definition of Negative Exponent 𝑓 𝑛 = 𝑔 1 𝑟 ∙ 𝑟 𝑛 Multiply So 𝑔 𝑛 = 𝑔 1 ∙ 𝑟 𝑛−1 can be written as 𝑓 𝑛 =𝑎∙ 𝑟 𝑛 . Most GEOMETRIC SEQUENCES represent an EXPONENTIAL FUNCTION

Geometric Sequences as Functions When would a geometric sequence NOT be an exponential function? When the common ratio is a NEGATIVE NUMBER!

Rewriting Sequences as Functions Rewrite the following sequences from your group activity as a function: 𝑔 𝑛 =45∙ 2 𝑛−1 𝑎 𝑛 =−4+2 𝑛−1 𝑔 𝑛 =−2∙ 3 𝑛−1 𝑎 𝑛 =4− 9 4 𝑛−1

Rewriting Functions as Sequences Rewrite the following functions as a sequence, using an explicit formula: 𝑓 𝑥 =5𝑥−17 𝑓 𝑥 =2∙ 3 𝑥