Unit 6 Guided Notes. Functions, Equations, and Graphs Standards: A.SSE.4, F.BF.1, F.BF.1a, F.BF.2, F.IF3, F.LE.2 Clio High School Algebra 2b

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1 Unit 6 Guided Notes Functions, Equations, and Graphs Standards: A.SSE.4, F.BF.1, F.BF.1a, F.BF.2, F.IF3, F.LE.2 Clio High School Algebra 2b Name: Period: Miss Seitz s tutoring: Tuesdays and Thursdays after school Website with all videos and resources Miss Kari Seitz Text: Classroom: kseitz@clioschools.org Concept # What we will be learning... Text Mathematical Patterns Write a function that describes the relationship between two quantities 9.1 Define an explicit and recursive expression of a function Arithmetic Sequences Define an arithmetic sequence and find the common difference between two terms Write recursive and explicit formulas for an arithmetic sequence Recognize that linear functions are a form of arithmetic sequence 9.2 Construct arithmetic sequences given a graph, description, or table Decide when real-world problems model an arithmetic sequence and write the equation to model the situation Geometric Sequences Define a geometric sequence and find the common ratio between two terms Write recursive and explicit formulas for a geometric sequence Recognize that exponential functions are a form of geometric sequence 9.3 Construct geometric sequences given a graph, description or table Decide when real-world problems model a geometric sequence and write the equation to model the situation Geometric Series Define a finite geometric series and find the common ratio Derive the formula for the sum of a finite geometric series, S n = a 1 ((1 r n )/(1 r)). 9.5 Express and calculate the sum of a finite geometric series Use the formula for the sum of a finite geometric series to solve real-world problems

2 Mathematical Patterns Text: 9.1 Write a function that describes the relationship between two quantities Define an explicit and recursive expression of a function Vocabulary: sequence, term of a sequence, explicit formula, recursive formula Definitions A S is an ordered list of numbers. Every number in the list is a T of the S We write them as a n where n denotes the position of that term in the sequence. Explicit Formula describes the nth term of a sequence using the number n Example 1: The sequence 2, 4, 6, 8 is shown at the right. Each term is t the value of the term number n, so you can write the explicit formula as a n = Example 2: A sequence has an explicit formula a n = 3n 2. What are the first ten terms of the sequence? Method 1: Write the formula. Substitute 1 for n and simplify. Substitute 2 for n and simplify. Continue until you reach the 10 th term. Method 2: Use a table to organize your work N a n = 3n - 2 a n Extension: What is the 40 th term of the sequence?

3 Example 3: Write an explicit formula for the sequence 4, 7, 10, 13, 16,... Think about the terms of the sequence and how they relate to the first term. a 1 = a 2 = + a 3 = + = + a 4 = + = + a 5 = + = + Now that we see a pattern, we can write the formula. 1.) Find the first 5 terms of the 2.) Find the 25 th term of the sequence: a n = 4n + 5 sequence from example 3. Recursive Formula relates each term to the one before it There are two parts to this kind of formula: 1.) I C : value of the first term (a 1 ) 2.) R F : relation between terms Example 4: The sequence 133, 130, 127, 124 shows a pattern where the next term is T L T the previous term. a 1 = a n = for n > 1 Write a recursive definition for the sequence. 3.) 1, 2, 6, 24, 120, 720,...

4 Arithmetic Sequences Text: 9.2 Write a function that describes the relationship between two quantities Define an explicit and recursive expression of a function Vocabulary: sequence, term of a sequence, explicit formula, recursive formula Definitions An A S is a sequence where the difference between consecutive terms is constant. Example 1: Is the sequence an arithmetic sequence? 3, 6, 9, 12, 15,... The difference between terms is called the C D. Example 2: Is the sequence an arithmetic sequence? 1, 4, 9, 16, 25,... Find the difference between the consecutive terms. Is there a common difference? The sequence. Find the difference between the consecutive terms. Is there a common difference? The sequence. Key Concept: Arithmetic Sequences A Recursive Definition for this sequence has two parts: a 1 = a Initial Condition a n = a n-1 + d, for n > 1 Recursive Formula An Explicit Definition for this sequence is a single formula: a n = a + (n 1) d, for n 1 Writing Explicit Formulas Example 3: Write the explicit formula. Then find the 43 rd term in each sequence. 12, 14, 16, 18,

5 Arithmetic Mean The A M is the average of two numbers x and y. It can be used to find the missing term in a sequence when the term before and term after are given. It is defined as. Example 4: Find the missing term of each arithmetic sequence. 23,, 49, 1.) Write the explicit formula. Then, find the 43 rd term. 2, 13, 24, 35, 2.) Find the missing term of the arithmetic sequence. 2,, 456,

6 Geometric Sequences Text: 9.3 Define a geometric sequence and find the common ratio between two terms Write recursive and explicit formulas for a geometric sequence Recognize that exponential functions are a form of geometric sequence Construct geometric sequences given a graph, description, or table Decide when real-world problems model a geometric sequence and write the equation to model the situation Vocabulary: geometric sequence, common ratio, geometric mean Definitions A G Sequence changes by the same amount for each term. This amount is called the C R. Identifying Geometric Sequences Example 1: Is the sequence a geometric sequence? If so, find the common ratio. 3, 9, 27, 81, Key Concept: Geometric Sequences A Recursive Definition for the sequence has two parts: a 1 = a Initial Condition a n = a n 1 r, for n > 1 Recursive Formula An Explicit Definition for this sequence is a single formula: a n = a 1 r n 1, for n 1 Writing Explicit Formulas Example 2: Write the explicit formula. Then generate the first 5 terms of the sequence. a 1 = 3, r = 2

7 Geometric Mean The G M of two positive numbers x and y is. Example 3: Find the missing term of each geometric sequence. 4,, 16,... 1.) Write the explicit formula. Then find the 10 th term in each sequence 3, 12, 48,... 2.) Find the missing term of each geometric sequence. 3,, 12,...

8 Geometric Series Text: 9.5 Define a finite geometric series and find the common ratio Derive the formula for the sum of a finite geometric series, S n = a 1 ((1 r n )/(1 r)) Express and calculate the sum of a finite geometric series Use the formula for the sum of a finite geometric series to solve real-world problems Vocabulary: geometric series, infinite, finite Definitions A G S is the sum of the terms of the geometric sequence. That means you take all of the terms in the sequence and add them up! F means there is an end to the sequence. I means the sequence goes on forever. Key Concept: Sum of a Finite Geometric Series The sum S n of a finite geometric series a 1 + a 1 r + a 1 r a 1 r n-1, r 1, is Where a 1 is the first term, r is the common ratio and n is the number of terms. Finding the Sum of a Finite Geometric Series Example 1: Evaluate each finite series for the specified number of terms ; n = 10 1.) Evaluate the finite series for the specified number of terms ; n = 15

9 Real World Application Example 2: This month, your friend deposits $400 to save for a vacation. She plans to deposit 10% more each successive month for the next 11 months. How much will she have saved after the 12 deposits?