Download 3(n – 1).

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,
+3
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, …,
+3 +3
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,
+3 +4
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, …,
+3 +4 +5

…
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
xn = 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 4th
term for
3, 8, 13, 18, 23, 28, 33, …
Arithmetic Sequence Rule
Write the rule and calculate the 4th 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:
xn = a + d(n-1)
xn = 3 + 5(n – 1)
so the 4th term is
x4 = 3 + 5(4-1) = 18
Evaluating Algebraic Expressions
Find the first four terms of the
sequence represented by the
expression xn = 4 + 3n.
Position (n)
4 + 3n
Term
1
2
3
4
4 + 3 ∙ 1 4 + 3 ∙2 4 + 3 ∙3 4 + 3 ∙4
7
10
13
16
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
xn = 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
20th term in the sequence.
Make a table that pairs each term’s
position with its value.
Position (n)
Term
1
2
3
…
20
∙3
∙3
∙3
∙3
∙3
3
6
9
…
60
Geometric Sequence Rule
Write an algebraic rule for the sequence 3, 6, 9, 12, …
then find the 20th 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:
xn = ar(n-1)
xn = (3)(3)(n-1)
so the 20th term is
x20 = (3)(3)(20-1)
Evaluating Algebraic Expressions
Find the first four terms of the
sequence represented by the
expression 3(n – 1).
Position (n)
3(n – 1)
Term
1
2
3
4
Evaluating Algebraic Expressions
Example 2: Find the first four terms
of the sequence represented by the
expression 3(n – 1).
Position (n)
1
2
3
4
3(n – 1)
3(1-1)
3(2-1)
3(3-1)
3(4-1)
Term
0
3
6
9