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How can exponential functions be identified through tables, graphs, and equations? How are the laws of exponents used to simplify and evaluate algebraic expressions? How can exponential functions be used to model real world data? What are geometric sequences and how are they related to exponential functions? ADD Multiply you _______ the Exponents Divide youSUBTRACT ________ the Exponents MULTIPLY the Exponents Power to a Power you ________ ONE Monomial has _______ term Binomial has ________ terms TWO THREE Trinomial has ________ terms > ONE terms Polynomial has ______ ADD the exponents To find the degree of a monomial you ____ LAREGEST To find the degree of a polynomial you use the __________ degree of the monomials. LIKE terms When adding polynomials you add _____ CHANGE SIGNS then When subtracting polynomials you _____________ LIKE add _______ terms. CFU3102.3.1; Recognize and extend arithmetic and geometric sequences. CLE 3102.3.1; Use algebraic thinking to analyze and generalize patterns. SPI 3102.2.1; Operate (add, subtract, multiply, divide, simplify, powers) with radicals and radical expressions including radicands involving rational numbers and algebraic expressions SPI 3102.3.1 Express a generalization of a pattern in various representations including algebraic and function notation. ½ ½ ½ ½ Can you multiply or divide Y by the same number each time? YES ½ + 6 + 6 + 6 Can you multiply or divide Y by the same number each time? No + 6 + 6 Sequence Set of numbers in a specific order Terms 0 8 16 24 32 Common Difference +8 Arithmetic Sequence Numerical Pattern that increases or decreases at a constant rate or value – Common Difference nth term = a1+(n-1)d 4th term = 0+(4-1)8 = 24 Identify Geometric Sequences A. Determine whether the sequence is arithmetic, geometric, or neither. Explain. 0, 8, 16, 24, 32, ... 0 8 8–0=8 16 24 16 – 8 = 8 24 – 16 = 8 32 32 – 24 = 8 Answer: The common difference is 8. So, the sequence is arithmetic. Identify Geometric Sequences B. Determine whether the sequence is arithmetic, geometric, or neither. Explain. 64, 48, 36, 27, ... 64 48 __ 3 = 4 64 48 ___ 36 __ 3 = 4 48 36 ___ 27 __ 3 = 4 36 27 ___ 3 , so the sequence is Answer: The common ratio is __ 4 geometric. A. Determine whether the sequence is arithmetic, geometric, or neither. 1, 7, 49, 343, ... A. arithmetic B. geometric C. neither Find Terms of Geometric Sequences A. Find the next three terms in the geometric sequence. 1, –8, 64, –512, ... Step 1 1 Find the common ratio. –8 64 –512 The common ratio is –8. __ = –8 –8 1 –512 64 ___ –8 4096 × (–8) = –8 –512 ______ 64 –32,768 × (–8) × (–8) = –8 262,144 Find Terms of Geometric Sequences B. Find the next three terms in the geometric sequence. 40, 20, 10, 5, .... Step 1 40 40 ___ 20 Find the common ratio. 20 = 10 __ 1 10 ___ 2 20 = 5 5 __ 1 ___ 2 10 = __ 1 2 1. The common ratio is __ 2 Answer: The next 3 terms in the sequence are __ 5 , and __ 5. 4 8 5 ,__ 2 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, a1 = 1. Now find the common ratio. 1 –2 4 –8 The common ratio is –2. –2 = –2 ___ 4 = –2 ___ –8 = –2 ___ 1 –2 4 an = a1rn – 1 Formula for the nth term an = 1(–2)n – 1 Answer: an = 1(–2)n – 1 a1 = 1 and r = –2 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. Compared to each previous hour, 80% of the weight remains. So, r = 0.80. 50, 40, 32, 25.6, 20.48,…. So after 1 hour, the sculpture weighs 40 pounds, 32 pounds after 2 hours, 25.6 pounds after 3 hours, and so forth. 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. C. B. D. Practice Page 581, 6 - 30 even*