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 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, … What are the next 3 terms in the sequence? 23, 27, 31 7, 11, 15, 19, 23, 27, 31 Answer: n + 4 (since you add 4 to generate each new term) Ex #2 49, 41, 33, 25, _____, _____, _____ circle arithmetic geometric neither Rule_______________________ *Yes, this is just subtraction; however, since arithmetic means adding, write is as addition. n + (−8)

Identify the pattern in the sequence as: arithmetic, geometric, or neither. 3, 6, 12, 24, … Answer: geometric You multiplied to generate each new term. Geometric 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, … What are the next 3 terms in the sequence? 48, 96, 192 3, 6, 12, 24, 48, 96, 192 Answer: 2n (since you multiplied by 2 to generate each new term) Ex #2 7203, 1029, 147, _____, _____, _____ circle arithmetic geometric neither Rule_______________________ * Yes, this is just division; however, since geometric means multiply, write is as multiplication. n

Describe the pattern in the sequence and identify the sequence as arithmetic, geometric, or neither. Other 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 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 changing …) (…it’s rising quickly … signs alternating … ) (…it’s falling quickly … signs not changing …) c. −99, −103, −107, −111 (…it’s falling slowly … signs not 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 Sequences a. 0.6, 1.86, 5.766, Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms. geometric – you’re multiplying by n , b. 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 , 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, , … Is it arithmetic, geometric, or neither? What’s the rule? List the next 2 terms. arithmetic – you’re adding n , 55.75, … (… number of decimal places increasing … ) (…subtract 1 st two terms, then the last 2 … same?) (…divide 1 st two terms, then the last 2 … same?) (…subtract 1 st two terms, then the last 2 … same?) (…divide 1 st two terms, then the last 2 … same?) (…subtract 1 st two terms, then the last 2 … same?)

Fractional Sequences a. 1, 1, 13, 3, Find the L east C ommon D enominator. ?, ?, ?, ?, … Rewrite each fraction with a new numerator and denominator. numerator and denominator. ● 6 = ● =24 ● 8 =8 8 ● ? =24 6 ●3 = 3 ●? = Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms. 5 arithmetic – you’re adding by 5 n , 28, 33 or 23, 7, b. 28, 7, 7, 7, … Find the L east C ommon D enominator. ?, ?, ?, ?, … Rewrite each fraction with a new numerator and denominator. numerator and denominator. ● 4 = ● =16 ● 16 = 16 ● ? =16 4 ●16= ● ? = Is it arithmetic, geometric, or neither? What’s the rule? List the next 3 terms. geometric – you’re multiplying by 0.25, or n or 1n 7, 7,

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 Sequences arithmetic n , 22.5, 27 arithmetic n + (−3) −8, −11, −14 arithmeticn , 10.5, 12.5 geometric −3n −243, 729, −2187 geometric −1n 4, −4, 4 geometric2n2n −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 More Practice with Sequences arithmetic n + 4 −60, −56, −52arithmeticn , 1.8, 2.1 neither add −2, then add −3, then −4, … −20, −27, −35geometric n n 6n6n, 16, 96 arithmeticn +, 4, arithmeticn +,, 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. 2 sodas ~ Later, you input 4 quarters, but the output is 2 sodas ? Functions function Is the machine doing its function correctly? function A relation is a function 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)} NOT Answer: It is NOT a function (an x−value, −3, repeats with a different y−value) Functions IS Answer: It IS a function (no x−values repeat) 2. {(2, 5), (4, –8), (3, 1), (6, −8), (–7, –9)} NOT It is NOT a function (an x−value, 1, repeats with a different y−value) IS It IS a function (no x−values repeat). IS It IS a function (an x−value, −4, repeats with the SAME y−value, 11) IS It IS a function (no x−values repeat)

Determine whether each graph is a function. Explain. If NO x−values repeat, it IS a function. Functions Use “vertical line test” to test for a function: 1. Hold a pencil vertically 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 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.

Functions There are different ways to show each part of a function. Let’s use the example of : T he effect of temperature 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. Functions 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 p 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 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 d + d Writing Functions As Variable Expressions

To 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 Table xyxy xy x y = 4x + 3 y (x,y) y = 4( ) + 3 (, ) y = 4( ) (, )

16 Graphing 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 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 Table xyxyxy x y = x – 2 y (x,y) y = ( ) – (, ) –2 –2 –1 0 –

18 Graphing 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 (y =) 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 ) ~ Plot the points, then draw your line. 1 2

20 Graphing Vertical (x = ) 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, ) ~ Plot the points, then draw your line. 1 2

common difference In an arithmetic sequence, you add the ________ to get each new term. 14, 3, −8,... +(−11)

common ratio In a geometric sequence, you multiply by the ________ to get each new term. 3, 21, 147,

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6n, geometric n + 12, arithmetic 7n, geometric 3n, geometric n + 1.1, arithmetic 3, 4, 5 …, neither 53, 58, , 24.6, , , , 1.6, , 3072, , 2020, 2037 Sequences