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Unit Four - Functions 8.F.1 Understand that a function is a rule that assigns exactly one output to each input. The graph of a function is the set of ordered pairs consisting of an input (x) and the corresponding output (y). 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
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Patterns and sequences We often need to spot a pattern in order to predict what will happen next. In math, the correct name for a pattern of numbers is called a SEQUENCE. Each number in a SEQUENCE is called a TERM. We can use the pattern to find additional terms in the sequence.
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Patterns and sequences For any pattern, it is important to try to spot what is happening before you can predict the next number. The first 2 or 3 numbers is rarely enough to show the full pattern - 4 or 5 numbers are best.
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Patterns and sequences Here are the first two numbers in a sequence: 1, 2, …… What’s do you think the next number might be?
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Patterns and sequences 1, 2, 4,…Who thought that the next number was 3? What number comes next?
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Patterns and sequences 1, 2, 4,…Who thought that the next number was 3? What number comes next? 1, 2, 4, 8, 16, …
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Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3
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Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3+ 3 X
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Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3+ 4
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Patterns and sequences Look at what is happening from 1 TERM to the next. See if that is what is happening for every TERM. 5, 8, 12, 17, 23, …, … + 3+ 4+ 5
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Patterns and sequences Now try this pattern – find the next three terms in the sequence. 3, 7, 11, 15, 19, …, …
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Patterns and sequences Now try this pattern – find the next three terms in the sequence. 3, 7, 11, 15, 19, …, … This is an example of an arithmetic sequence. Each term differs from the next by a fixed number, called the common difference.
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Arithmetic Sequences Arithmetic Sequences can be written as a rule or expression. The rule gives the nth term, where n is the term’s position in the sequence.
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Arithmetic Sequence Rule x n = a + d(n-1) where: a is the first term d is the common difference (We use "n-1" because d is not used in the 1st term).
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Arithmetic Sequence Rule Write the rule and calculate the 4 th term for 3, 8, 13, 18, 23, 28, 33, …
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Arithmetic Sequence Rule Write the rule and calculate the 4 th term for 3, 8, 13, 18, 23, 28, 33, … The first term in the sequence is 3. The sequence has a difference of 5 between each number.
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Arithmetic Sequence Rule a = 3 (the first term) d = 5 (the common difference) The rule for this sequence is: x n = a + d(n-1) x n = 3 + 5(n – 1) so the 4 th term is x 4 = 3 + 5(4-1) = 18
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Evaluating Algebraic Expressions Find the first four terms of the sequence represented by the expression x n = 4 + 3n. Position (n)1234 4 + 3n4 + 3 ∙ 14 + 3 ∙24 + 3 ∙34 + 3 ∙4 Term7101316
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Patterns and sequences Now try this pattern – find the next two terms in the sequence. 2, 6, 18, 54, …, …
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Patterns and sequences 2, 6, 18, 54, 162, 486, … This is an example of a geometric sequence. Each term in the sequence is found by multiplying the previous term by a fixed number called the common ratio.
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Geometic Sequence Rule x n = ar (n-1) where: a is the first term r is the common ratio (Again, we use "n-1" because r is not used in the 1st term).
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Geometric Sequence Rule Write an algebraic rule for the sequence 3, 6, 9, 12, … then find the 20 th term in the sequence. Make a table that pairs each term’s position with its value. Position (n)123…20 ∙ 3 Term369…60
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Geometric Sequence Rule Write an algebraic rule for the sequence 3, 6, 9, 12, … then find the 20 th term in the sequence. The first term in the sequence is 3. The sequence has a ratio of 3 between each number.
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Geometic Sequence Rule a = 3 (the first term) r = 3 (the common ratio) The rule for this sequence is: x n = ar (n-1) x n = (3)(3) (n-1) so the 20 th term is x 20 = (3)(3) (20-1)
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Evaluating Algebraic Expressions Find the first four terms of the sequence represented by the expression 3(n – 1). Position (n)1234 3(n – 1) Term
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Evaluating Algebraic Expressions Example 2: Find the first four terms of the sequence represented by the expression 3(n – 1). Position (n)1234 3(n – 1)3(1-1)3(2-1)3(3-1)3(4-1) Term0369
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