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).

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.

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.

Patterns and sequences Here are the first two numbers in a sequence: 1, 2, …… What’s do you think the next number might be?

Patterns and sequences 1, 2, 4,…Who thought that the next number was 3? What number comes next?

Patterns and sequences 1, 2, 4,…Who thought that the next number was 3? What number comes next? 1, 2, 4, 8, 16, …

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

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, …, … X

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, …, …

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, …, … 

Patterns and sequences Now try this pattern – find the next three terms in the sequence. 3, 7, 11, 15, 19, …, …

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.

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.

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).

Arithmetic Sequence Rule Write the rule and calculate the 4 th term for 3, 8, 13, 18, 23, 28, 33, …

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.

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

Evaluating Algebraic Expressions Find the first four terms of the sequence represented by the expression x n = 4 + 3n. Position (n) n4 + 3 ∙ ∙ ∙ ∙4 Term

Patterns and sequences Now try this pattern – find the next two terms in the sequence. 2, 6, 18, 54, …, …

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.

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).

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

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.

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)

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

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