Identify the pattern in the sequence as: arithmetic, geometric, or neither. 7, 11, 15, 19, … Answer: arithmetic You added to generate each new term. Arithmetic.

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


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


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, …


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 , …


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,..


geometric – you’re multiplying by 4 4n 409.6, 1638.4, …


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?)






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


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


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








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, …









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

30

31

32

33

34

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

Seq

uen

ces

Seq

uen

ces

Identify the pattern in the sequence as: arithmetic, geometric, or neither. 7, 11, 15, 19, … Answer: arithmetic You added to generate each new term. Arithmetic.

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