describe the pattern in the sequence and identify the sequence as

Describe the pattern in the sequence and identify the sequence as

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

Answer: n + 4 (since you add 4 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 multiply by 2 to

generate each new term)

3, 6, 12, 24, … Answer: geometric

You multiplied to generate each new term.

*NOTE: these answers are not acceptable:

2•n , n•2 , 2 × n , n × 2 , n2



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 a. 17 , 9 , 1 , -7 , …

Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.

arithmetic – you’re adding -8* n + (-8) -15, -23, -31, ...

b. -4 , 12 , -36 , 108 , …


geometric – you’re multiplying by -3 -3n 324, -972, 2916, ...

*Note: If you’re thinking, “Isn’t this just subtraction?”, you’re right; however, since arithmetic sequences mean adding, we’ll write them as

addition.

c. -37, -32, -27, -22, …


arithmetic – you’re adding +5 n + 5 -17, -12, -7, … d. -1, -7, -49, -343, …


geometric – you’re multiplying by +7 7n -2401, -16807, -117649

Decimal Number SequencesDecimal Number Sequences a. 3 , 1.5 , 0.75 , 0.375 …


geometric – you’re multiplying by 0.5 0.5n or .5n 0.1875, 0.09375, 0.046875

b. 4.7 , 7 , 9.3 , 11.6 , …Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms.

arithmetic – you’re adding 2.3 n + 2.3 13.9, 16.2, 18.5, …

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 3 terms.

geometric – you’re multiplying by 4 4n 409.6, 1638.4, 6553.6


arithmetic – you’re adding 10.25 n + 10.25 45.5, 55.75, 66, …

Fractional SequencesFractional Sequences a. 28, 7, , , …


geometric – you’re multiplying by n or , ,


arithmetic – you’re adding n + , , 18

4

7

16

7

4

14

1

4

n

1024

7

256

7

64

7

*Note: If you’re thinking, “Isn’t this just division?”, you’re right; however, since geometric sequences mean multiplying, we’ll write them as

multiplication.

>>>> remember: ÷÷4 is the same as •• <<<4

1

b. -2 , , , 8 , …3

4

3

14

3

103

10

3

34

3

44

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

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

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.

Remember, if nono x-values repeat, it IS a function.

FunctionsFunctions

Here’s a trick to find out:1. Hold a pencil vertically ...2. Then, slide it across the curve. * Does the pencil ever hit the curve more than once?

The pencil does NOT hit the curve more than once, so it IS a function.

It PASSES the vertical line test.

Remember, if nono x-values repeat, it IS a function.

Here’s a trick to find out:1. Hold a pencil vertically ...2. Then, slide it across the curve. * Does the pencil ever hit the curve more than once?

The pencil DOES hit the curve more than once, so it is NOT a function.

It FAILS the vertical line test.

If it DOES hit the curve more than once, it is

NOT a function.

If it does NOT hit the curve

more than once, it IS a

function.

If it DOES hit the curve more than once, it is

NOT a function.

If it does NOT hit the curve

more than once, it IS 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 yy

(xx,yy)

y = 4( ) + 3

( , )

y = 4( ) + 3

y = 4( ) + 3

00 00 001122

11 1122 22

( , )

( ,

)

33 33771111

771111

17

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

( ,

)

19

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)

20

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.

( 1, 4 )( 2, 4 )

~ Plot the points, then draw your line.

21

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 , 1)(–7 , 2)

~ Plot the points, then draw your line.

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

describe the pattern in the sequence and identify the sequence as

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