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Identify the pattern in the sequence as:
arithmetic, geometric, or neither. arithmetic, geometric, or neither.
7, 11, 15, 19, …Answer: arithmetic
You added to generate each new term.
Arithmetic SequencesArithmetic Sequences
What is the rule used to generate new terms in the sequence?
Write it as a variable expression, and use n to represent the last number given.
7, 11, 15, 19, …
b. What are the next 3 terms in the sequence?
7, 11, 15, 19, 23, 27, 31 23, 27, 31
Answer: n + 4 (since you add 4 to
generate each new term)
Ex #2 49, 41, 33, 25 , _____ , _____ , _____
circlecircle
arithmetic geometric arithmetic geometric neitherneither
RuleRule
______________________________________________
*Yes, this is just subtraction; however, since arithmetic means adding, write is as addition.
nn+ (−8)+ (−8) 17 9 1
Identify the pattern in the sequence as:
arithmetic, geometric, or neither. arithmetic, geometric, or neither.
3, 6, 12, 24, …Answer: geometric
You multiplied to generate each new term.
Geometric SequencesGeometric Sequences
What is the rule used to generate new terms in the sequence?
Write it as a variable expression, and use n to represent the last number given.
3, 6, 12, 24, …
b. What are the next 3 terms in the sequence?
3, 6, 12, 24, 48 , 96 , 192 48 , 96 , 192
Answer: 2n (since you multiplied by 2 to
generate each new term)
Ex #2 7203, 1029, 147, _____ , _____ , _____
circlecircle
arithmetic geometric arithmetic geometric neitherneither
RuleRule
______________________________________________
*Yes, this is just division; however, since geometric means multiply, write is as multiplication.
nn 21 3 37
17
Describe the pattern in the sequence and identify the sequence as
arithmetic, geometric, or neither.
Other SequencesOther Sequences
What is the rule used to generate new terms in the sequence?
Since the pattern is neither arithmetic nor geometric, you can state the rule in words.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
b.
What are the next 3 terms in the sequence?
Answer: You add the last 2 terms together to generate each new term)
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …Answer: neither
There’s a pattern, but you’re neither adding nor multiplying by the same number.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 , 89 , 144 , 233 , 377
=
Negative Number SequencesNegative Number Sequences
d. −4 , 12 , −36 , 108 , …
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
geometric – you’re multiplying by 7 7n −2401, −16,807 , −117,649
a. −37, −32, −27, −22, … Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
arithmetic – you’re adding +5 n + 5 −17, −12, −7, …b. −1, −7, −49, −343, …
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
geometric – you’re multiplying by −3 −3n −324, 972, −2916
(…it’s rising slowly … signs not (…it’s rising slowly … signs not changing …)changing …)
(…it’s rising quickly … signs alternating … )(…it’s rising quickly … signs alternating … )
(…it’s falling quickly … signs not changing …)(…it’s falling quickly … signs not changing …)
c. −99, −103, −107, −111 (…it’s falling slowly … signs not (…it’s falling slowly … signs not changing …)changing …)
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.arithmetic – you’re adding by −4 n + −4 −115, −119, −123
Decimal Number SequencesDecimal Number Sequencesa. 0.6 , 1.86 , 5.766 , 17.8746Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms.
geometric – you’re multiplying by 3.1 3.1n 55.41126, 171.774906b. 4.7 , 7 , 9.3 , 11.6 , …
Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms.
arithmetic – you’re adding 2.3 n + 2.3 13.9, 16.2, …c. 4.5, 14.75, 25, 35.25,..
d. 1.6, 6.4, 25.6, 102.4,..
Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms.
geometric – you’re multiplying by 4 4n 409.6, 1638.4, …
Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms.
arithmetic – you’re adding 10.25 n + 10.25 45.5, 55.75, …
(… number of decimal places increasing … )(… number of decimal places increasing … )
(…subtract 1(…subtract 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
(…divide 1(…divide 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
(…divide 1(…divide 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
(…divide 1(…divide 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
(…subtract 1(…subtract 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
(…divide 1(…divide 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
(…subtract 1(…subtract 1stst two terms, then the last 2 … same?) two terms, then the last 2 … same?)
Fractional SequencesFractional Sequencesa. 1, 1, 13, 3,
8 3 24 4
1. Find the 1. Find the LLeast east CCommon ommon
DDenominator. enominator. ? , ? , ? , ? , … 8 3 24 4
2.2. Rewrite each fraction with a Rewrite each fraction with a new new
numeratornumerator and and denominatordenominator..
● ● 6 =6 =
● ● ==2424
● ● 8 =8 =88
88 ● ● ?? = =2424 66
●●3 = 3 = 33
●●?? = =242433
1818
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
arithmetic – you’re adding by 55 n + 5 23 , 28 , 33 or 23 , 7 , 11 24 24 24 24 24 24 6 8
b. 28 , 7 , 7 , 7 , …
4 16
1. Find the 1. Find the LLeast east CCommon ommon
DDenominator. enominator. ? , ? , ? , ? , … 4 16
2.2. Rewrite each fraction with a Rewrite each fraction with a new new
numeratornumerator and and denominatordenominator..
● ● 44 = =
● ● ==1616
● 16 =
1616 ● ● ?? = =1616 44
●16=
● ● ?? = =1616161611 11
448448 112112 2828
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.
geometric – you’re multiplying by 0.25, or 1 0.25n or 1n 7 , 7 , 7 4 4 64 256 512
0, 4.5, 9, 13.5, … ‒3, ‒ 6, ‒12, ‒24, ‒48, . . .
1, ‒3, 9, ‒27, 81, . . . 1, 2, 1, 2, 1, . . .
‒4, 4, ‒4, 4, ‒4, . . . 0.5, 2.5, 4.5, 6.5, …
7, 4, 1, ‒2, ‒5, . . . ‒5, 10, ‒20, 40, ‒80, . . .
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Practice with SequencesPractice with Sequences
arithmetic n + 4.5 18, 22.5, 27
arithmetic n + (−3) −8, −11, −14
arithmetic n + 2 8.5, 10.5, 12.5
geometric −3n −243, 729, −2187
geometric −1n 4, −4, 4
geometric 2n −96, −192, −384
geometric −2n 160, −320, 640
neither add 1, then add −1
2, 1, 2
0, ‒2, ‒5, ‒9, ‒14, . . . 81, 27, 9, 3, 1, . . .
‒80, ‒76, ‒72, ‒68, ‒64, . . . 0.3, 0.6, 0.9, 1.2, …
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms
More Practice with SequencesMore Practice with Sequences
arithmetic n + 4 −60, −56, −52 arithmetic n + 0.3 1.5, 1.8, 2.1
neither add −2, then add −3, then
−4, …
−20, −27, −35 geometric n3
1
3
1
9
1
27
1
geometric n8
1
512
5
4096
5
32768
5
geometric 6n3
8 , 16, 96
arithmetic n + 3
2
3
14
3
10, 4,
arithmetic n + 8
1
8
7
4
3, , 1
A Coke machine charges $1.00 for a soda. ~ If your input is 1 quarter, your output will be 0 sodas. ~ If your input is 2 quarters, your output will be 0 sodas. ~ If your input is 4 quarters, your output will be 1 soda.
~ Later, you input 4 quarters, but the output is 2 2 sodassodas?
FunctionsFunctions
1 02 04 14 2
Is the machine doing its functionfunction correctly?
Is the machine doing its functionfunction correctly?
A relation is a functionfunction when:
~ No inputs repeat.
or
~ If an input repeats, it’s always paired with the same output.
Determine whether the relation is a function.1. {(–3, –4), (–1, –5), (0, 6), (–3, 9), (2, 7)}Answer: It is NOTNOT a function (an x−value, −3, repeats with a
different y−value)
FunctionsFunctions
3. 5. 6.
4.
Answer: It ISIS a function (no x−values repeat)
2. {(2, 5), (4, –8), (3, 1), (6, −8), (–7, –9)}
It is NOTNOT a function (an x−value, 1, repeats with a different y−value)
It ISIS a function (no x−values repeat).
It ISIS a function (an x−value, −4, repeats with the SAME
y−value, 11)
It ISIS a function (no
x−values repeat)
Determine whether each graph is a function. Explain.
If NO x−values repeat, it IS a function.
FunctionsFunctions
Use “vertical line test” to test for a function:1. Hold a pencil vertically ...
2. Then, slide it across the curve. *Does the pencil ever hit the curve TWICE?
The pencil hits the curve ONCE, so it PASSES the vertical line test.
It IS a function.
If NO x−values repeat, it IS a function.Use “vertical line test” to test for a function:1. Hold a pencil vertically ...
2. Then, slide it across the curve. *Does the pencil ever hit the curve TWICE?
The pencil hits the curve TWICE, so it FAILS the vertical line test.
It is NOT a function.
If the pencil hits the
curve ONCE, it IS a
function.
If the pencil hits the curve
TWICE, it is NOT a
function.
If the pencil hits the
curve ONCE, it IS a
function.
If the pencil hits the curve
TWICE, it is NOT a
function.
FunctionsFunctionsThere are different ways to show each part of a function. Let’s use the example of: The effect of The effect of temperature on cricket chirpstemperature on cricket chirps
Which variable causes the
change?
Which variable responds to the
change?Which letter is listed first in an ordered pair?
Which letter is listed second in an ordered pair?
This is the list of all input (x) values.
This is the list of all output (y) values.
This is what goes in. This is what comes out.
Conclusion: As temperature increases, cricket chirps increase.(Summary):
texts per week average quiz score
a. What is the input? output?
A teacher displays the results of her survey of
her students.
FunctionsFunctions
b. What is the independent variable? dependent variable?
{10, 25, 100, 200} {81, 87, 94}
texts per week avg quiz score
c. What are all the x−values? y−values?
{10, 25, 100, 200} {81, 87, 94}
d. What’s the domain? range?
Matt is a manager at Dominos. He earns a salary of $500/week, but he also gets $0.75 for every
pizza he sells. Write a variable expression you could
use to find his total weekly pay.salary + pay per pizza = total weekly pay
500 + 0.75 • p 500 + 0.75p
Ned sells tandem skydives. He makes $1000 for a full plane of jumpers, but he has to pay the
pilot $25 per jumper. Write a variable expression you could use to find his total pay for every full
plane.Ned’s pay ‒ pay per jumper =
total pay 1000 ‒ 25 • j
1000 ‒ 25j
Adam drives a truck, and his mileage chart is above.
Write a variable expression you could use to find his total amount
of gas he has in his tank?gas he started with – gas per mile = gas
remaining 35.1 − 0.6 • m
35.1 ‒ 0.6m
Lambert is running a
food donation
drive, and the results are to the
right.Write a variable
expression you could use
to find his total pounds
of food donated?
starting food + food per day
97 + 2 • d3
1
6
1
+ d
Writing Functions As Variable ExpressionsWriting Functions As Variable Expressions
To graph a functionTo graph a function~ Step 1: Pick a value for x ( I recommend “0”), then ... * Write “0” under “x”, ... * ... re−write your equation, then plug in “0” for x, then ... * ... plug in “0” for the x−value of the ordered pair.~ Step 2: To figure out the y−value, * Use order of operations to evaluate the expression. The “answer” is your y−value, so ...
> write it under “y”, ... > ... then plug it in for the y−value of your ordered pair.
Completing a Function TableCompleting a Function Table
xx yy = 4xx + 3 y y
(xx,yy)
y = 4( ) + 3
( , )
y = 4( ) + 3
y = 4( ) + 3
00 00 0011
2211 11
22 22( , )
( ,
)
33 33771111
771111
16
Graphing Functions with Ordered PairsGraphing Functions with Ordered Pairs
Plot all three ordered pairs from your function table
If they all line up,~ get a ruler, then ...~ draw a straight line through all 3 points.If they don’t line up,~ choose a new x−value~ plug it in your function table~ plot your new point (hopefully, they line up)
To graph a functionTo graph a function~ Step 1: Pick a value for x ( I recommend “0”), then ... * Write “0” under “x”, ... * ... re−write your equation, then plug in “0” for x, then ... * ... plug in “0” for the x−value of the ordered pair.~ Step 2: To figure out the y−value, * Use order of operations to evaluate the expression. The “answer” is your y−value, so ...
> write it under “y”, ... > ... then plug it in for the y−value of your ordered pair.
Completing a Function TableCompleting a Function Table
xx yy = xx – 2 yy
(xx,yy)
y = ( ) – 2
y = ( ) – 2
y = ( ) – 2
00 00 004488
44 4488 88
( , )
( ,
)
––22 ––22––1100
––1100
14
14
14
14
( ,
)
18
Graphing Functions with Ordered PairsGraphing Functions with Ordered Pairs
Plot all three ordered pairs from your function table
If they all line up,~ get a ruler, then ...~ draw a straight line through all 3 points.If they don’t line up,~ choose a new x−value~ plug it in your function table~ plot your new point (hopefully, they line up)
19
Graphing Horizontal (Graphing Horizontal (y =y =) Lines.) Lines.
Graph y = 4~ Write an ordered pair with any x−value.( 0 , )~ The y−value is 4. * Why? Because the original equation is y = 4.
4
~ Pick another x−value. The y−value will be 4.( , 4 )
( , 4 )~ Plot the points, then draw your line.
12
20
Graphing Vertical (Graphing Vertical (x =x = ) Lines. ) Lines.
Graph x = –7~ Write an ordered pair with any y−value.( , 0 )~ The x−value is –7. * Why? Because the original equation is x = –7.
–7
~ Pick another y−value. The x−value is –7. (–7 , )
(–7 , )~ Plot the points, then draw your line.
12
common differenc
e
In an arithmetic sequence, you add the ________ to get
each new term.
14, 3, −8, ...
+(−11) +(−11)
common ratio
In a geometric sequence, you multiply by the ________ to get each new term.
3, 21, 147,...
•7 •7
66nn , geometric , geometric n n + 12 , arithmetic+ 12 , arithmetic
77nn , geometric , geometric 33nn , geometric , geometric
n n + 1.1 , arithmetic+ 1.1 , arithmetic ••3, •4, •5 … , neither3, •4, •5 … , neither
53, 58, 63 53, 58, 63 20.6, 24.6, 28.620.6, 24.6, 28.6 204.8, 1638.4, 13107.2204.8, 1638.4, 13107.2
1.3, 1.6, 1.91.3, 1.6, 1.9768, 3072, 12288768, 3072, 12288
125
1,
25
1,
5
1
2013, 2020, 20372013, 2020, 2037
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