Splash Screen. CCSS Content Standards F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model.

Slides:

Advertisements

Similar presentations

Splash Screen. Then/Now I CAN identify and generate geometric sequences and relate them to exponential functions. Learning Target.

Advertisements

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–3) CCSS Then/Now New Vocabulary Example 1: Parallel Line Through a Given Point Example 2:

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 3–4) CCSS Then/Now New Vocabulary Key Concept: Arithmetic Sequence Example 1:Identify Arithmetic.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–2) CCSS Then/Now New Vocabulary Key Concept: Point-Slope Form Example 1:Write and Graph an.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 3–5) CCSS Then/Now Key Concept: Proportional Relationship Example 1:Real-World Example: Proportional.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 7–7) CCSS Then/Now Example 1:Use a Recursive Formula Key Concept: Writing Recursive Formulas.

Splash Screen. CCSS Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 7–2) CCSS Then/Now New Vocabulary Key Concept: b Example 1: Radical and Exponential Forms Key.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 7–4) CCSS Then/Now New Vocabulary Key Concept: Exponential Function Example 1:Graph with a >

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–4) CCSS Then/Now New Vocabulary Concept Summary: Scatter Plot Example 1:Real-World Example:

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–1) CCSS Then/Now New Vocabulary Example 1:Write an Equation Given the Slope and a Point Example.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 7–5) CCSS Then/Now New Vocabulary Key Concept: Equation for Exponential Growth Example 1:Real-World.

What are two types of Sequences?

Geometric Sequences as Exponential Functions

Over Lesson 7–6 5-Minute Check 1 The number of people who carry cell phones increases by 29% each year. In 2002, there were 180 million cell phone users.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 7–2) CCSS Then/Now New Vocabulary Key Concept: b Example 1: Radical and Exponential Forms Key.

Homework Questions. Geometric Sequences In a geometric sequence, the ratio between consecutive terms is constant. This ratio is called the common ratio.

COMMON CORE STANDARDS for MATHEMATICS FUNCTIONS: INTERPRETING FUNCTIONS (F-IF) F-IF3. Recognize that sequences are functions, sometimes defined recursively.

$100 $200 $300 $400 $500 $200 $300 $400 $500 Exponents Scientific Notation Exponential Growth and Decay Properties of exponents Geometry Sequences.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–2) CCSS Then/Now New Vocabulary Key Concept: Point-Slope Form Example 1:Write and Graph an.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–4) CCSS Then/Now New Vocabulary Concept Summary: Scatter Plot Example 1:Real-World Example:

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–3) CCSS Then/Now New Vocabulary Example 1: Parallel Line Through a Given Point Example 2:

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 3–4) CCSS Then/Now New Vocabulary Key Concept: Arithmetic Sequence Example 1:Identify Arithmetic.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–1) CCSS Then/Now New Vocabulary Example 1:Write an Equation Given the Slope and a Point Example.

Over Lesson 3–2 5-Minute Check 5 A.264 B.222 C.153 D.134 The equation P = 3000 – 22.5n represents the amount of profit P a catering company earns depending.

Arithmetic and Geometric Sequences. Determine whether each sequence is arithmetic, geometric, or neither. Explain your reasoning. 1. 7, 13, 19, 25, …2.

Name the next number in the sequence: a) 1, 3, 9, 27, … b) 1, 5, 25, 125, …

+ 8.4 – Geometric Sequences. + Geometric Sequences A sequence is a sequence in which each term after the first is found by the previous term by a constant.

Splash Screen. Lesson Menu Five-Minute Check (over Chapter 6) CCSS Then/Now New Vocabulary Example 1: Identify Monomials Key Concept: Product of Powers.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 7–5) CCSS Then/Now New Vocabulary Key Concept: Equation for Exponential Growth Example 1:Real-World.

Over Lesson 7–7 5-Minute Check 1 Describe the sequence as arithmetic, geometric or neither: 1, 4, 9, 16, …? Describe the sequence as arithmetic, geometric,

A.y = 2x + 5 B.y = 2x – 5 C.y + 5 = 2(x – 5) D.y = 2(x – 5) What is the point-slope form of an equation for a line that passes through the point (5, –5)

How can exponential functions be identified through tables, graphs, and equations? How are the laws of exponents used to simplify and evaluate algebraic.

3.5 Arithmetic Sequences as Linear Functions

Five-Minute Check (over Lesson 3–2) Mathematical Practices Then/Now

7-Chapter Notes Algebra 1.

11.3 – Geometric Sequences.

Homework Questions.

Unit 3: Linear and Exponential Functions

Presentation transcript:

Splash Screen

CCSS Content Standards F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Mathematical Practices 7 Look for and make use of structure. Common Core State Standards © Copyright National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

Then/Now You related arithmetic sequences to linear functions. Identify and generate geometric sequences. Relate geometric sequences to exponential functions.

Example 1 Identify Geometric Sequences A. Determine whether the sequence is arithmetic, geometric, or neither. Explain. 0, 8, 16, 24, 32, – 0 = 8 Answer: The common difference is 8. So, the sequence is arithmetic. 16 – 8 = 824 – 16 = 832 – 24 = 8

Example 1 Identify Geometric Sequences B. Determine whether the sequence is arithmetic, geometric, or neither. Explain. 64, 48, 36, 27, Answer: The common ratio is, so the sequence is geometric. __ ___ = __ 3 4 ___ = __ 3 4 ___ =

Example 1 A.arithmetic B.geometric C.neither A. Determine whether the sequence is arithmetic, geometric, or neither. 1, 7, 49, 343,...

Example 1 B. Determine whether the sequence is arithmetic, geometric, or neither. 1, 2, 4, 14, 54,... A.arithmetic B.geometric C.neither

Example 2 Find Terms of Geometric Sequences A. Find the next three terms in the geometric sequence. 1, –8, 64, –512,... 1 –8 64 –512 The common ratio is –8. = –8 __ 1 –8 ___ 64 –8 = –8 ______ – Step 1Find the common ratio.

Example 2 Find Terms of Geometric Sequences Step 2Multiply each term by the common ratio to find the next three terms. 262,144 × (–8) Answer: –32, –512

Example 2 Find Terms of Geometric Sequences Step 2Multiply each term by the common ratio to find the next three terms. 262,144 × (–8) Answer: The next 3 terms in the sequence are 4096; –32,768; and 262,144. –32, –512

Example 2 Find Terms of Geometric Sequences B. Find the next three terms in the geometric sequence. 40, 20, 10, 5, Step 1Find the common ratio. = __ 1 2 ___ = __ 1 2 ___ = __ 1 2 ___ 5 10 The common ratio is. __ 1 2

Example 2 Find Terms of Geometric Sequences Step 2Multiply each term by the common ratio to find the next three terms. 5 __ × 1 2 × 1 2 × 1 2 Answer:

Example 2 Find Terms of Geometric Sequences Step 2Multiply each term by the common ratio to find the next three terms. 5 __ × 1 2 × 1 2 × 1 2 Answer: The next 3 terms in the sequence are, __ , and. __ 5 8

Concept

Example 3 Find the nth Term of a Geometric Sequence A. Write an equation for the nth term of the geometric sequence 1, –2, 4, –8,.... The first term of the sequence is 1. So, a 1 = 1. Now find the common ratio. 1 –2 4 –8 = –2 ___ –2 1 = –2 ___ 4 –2 = –2 ___ –8 4 a n = a 1 r n – 1 Formula for the nth term a n = 1(–2) n – 1 a 1 = 1 and r = –2 The common ratio is –2. Answer: a n = 1(–2) n – 1

Example 3 Find the nth Term of a Geometric Sequence B. Find the 12 th term of the sequence. 1, –2, 4, –8,.... a n = a 1 r n – 1 Formula for the nth term a 12 = 1(–2) 12 – 1 For the nth term, n = 12. = 1(–2) 11 Simplify. = 1(–2048)(–2) 11 = –2048 = –2048Multiply. Answer: The 12 th term of the sequence is –2048.

Example 3 A.768 B.–3072 C.12,288 D.–49,152 B. Find the 7th term of this sequence using the equation a n = 3(–4) n – 1.

Example 4 Graph a Geometric Sequence ART A 50-pound ice sculpture is melting at a rate in which 80% of its weight remains each hour. Draw a graph to represent how many pounds of the sculpture is left at each hour.

Example 4 Graph a Geometric Sequence Answer:

Example 4 SOCCER A soccer tournament begins with 32 teams in the first round. In each of the following rounds, one half of the teams are left to compete, until only one team remains. Draw a graph to represent how many teams are left to compete in each round. A.B. C.D.

Example 4 SOCCER A soccer tournament begins with 32 teams in the first round. In each of the following rounds, one half of the teams are left to compete, until only one team remains. Draw a graph to represent how many teams are left to compete in each round. A.B. C.D.