Algebra 1 Unit 2

Algebra 1 Unit 2

Notes & Assignments

Mrs. Cisneros, Rm 715

Name______________________Per___

Unit 2 Vocabulary

Match each word with its definition

_____1. Linear

_____2. Slope

_____3. Constant

_____4. Scatter Plot

_____5. Relation

_____6. Substitute

_____7. Line of Best Fit

_____8. Correlation

_____9. Expression

_____10. Equation

______11. Simplify

______12. Solve

______13. Inequality

______14. Function

______15. Term

______16. Commutative Property

______17. Evaluate

______18. Slope – Intercept Form

______19. Distributive Property

______20. Point – Slope Form

______21. Variable

______22. Rate of Change

a. A symbol or combination of symbolsrepresenting a value or relation

b. An expression asserting the equality oftwo quantities

c. To work out the answer or solution toan equation

d. To determine or calculate thenumerical value of an expression

e. To break down or make lesscomplicated or complex

f. A statement that two quantities areunequal with infinite solutions

g. Each member of an expressionseparated by + or – signs

h. An unknown value represented by aletter or symbol

i. A quantity assumed to be unchangedthroughout a given discussion

j. A relation between two sets involvingindependent and dependent variables

k. A ratio that describes, on average, howmuch one quantity changes withrespect to a change in anotherquantity

l. The ratio of the change in Y and thechange in X (m)

m. The property that associates twoquantities in a definite order

n. Extended or arranged in a lineo. Y = mx + bp. Y – y1 = m (X – X1)q. A graphic representation of bivariate

data on a coordinate planer. A line representing the closest data

pointss. The degree to which two or more

attributes show a tendency to varytogether

t. To replace an element with anotherelement of equal value

u. Multiplication over additionv. A + B = B + A or ab=ba

1

Name___________________________ Per____

Piggies & Pools Activity

1. My little sister, Savannah, is three years old. She has a piggy bank that she wants to fill. Shestarted with five pennies and each day when I come home from school, she is excited when Igive her three pennies that are left over from my lunch money. Use a table, a graph, and anequation to create a mathematical model for the number of pennies in the piggy bank on day n.

2. Our family has a small pool for relaxing in the summer that holds 1500 gallons of water. Idecided to fill the pool for the summer. When I had 5 gallons of water in the pool, I decidedthat I didn’t want to stand outside and watch the pool fill, so I had to figure out how long itwould take so that I could leave, but come back to turn off the water at the right time. Ichecked the flow on the hose and found that it was filling the pool at a rate of 2 gallons everyminute. Use a table, a graph, and an equation to create a mathematical model for the numberof gallons of water in the pool at t minutes.

3. Compare problems 1 and 2. What similarities do you see? What differences do you notice?

2

New Vocabulary 1. Domain:________________________________________________________________________________________________

__________________________________________________________________________________________________________

2. Discrete Function:_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

3. Continuous Function:_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Problem Sets READY Predict the next 2 terms in the sequence. Justify your answer. 1. 4, -‐20, 100, -‐500, … 2. 3, 5, 8, 12, …

3. 64, 48, 36, 27, … 4. 1.5, 0.75, 0, -‐0.75, …

5. 40, 10, 52, 58, … 6. 1, 11, 111, 1111, …

7. -‐3.6, -‐5.4, -‐8.1, -‐12.15, … 8. -‐64, -‐47, -‐30, -‐13, …

SET Identify whether the following statements represent a discrete or a continuous relationship. 9. The hair on your head grows ½ inch per month.

10. For every ton of paper that is recycles, 17 trees are saved.

11. Approximately 3.24 billion gallons of water flow over Niagara Falls daily.

12. The average person laughs 15 times per day.

13. The city of Buenos Aires adds 6,000 tons of trash to its landfills every day.

14. During the Great Depression, stock market prices fell 75%.

Adapted from the Mathematics Vision Project. Licensed under the Creative Commons Attribution CC BY 4.0. mathematicsvisionproject.org

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GO (HW)Topic:Solvingone-stepequationsEitherfindorusetheunitrateforeachofthequestionsbelow.

16. Applesareonsaleatthemarket4poundsfor$2.00.Whatistheprice(incents)foronepound?

17. Threeapplesweighaboutapound.Abouthowmuchwouldoneapplecost?(Roundtothenearestcent.)

18. Onedozeneggscost$1.98.Howmuchdoes1eggcost?(Roundtothenearestcent.)

19. Onedozeneggscost$1.98.Ifthechargeattheregisterforonlyeggs,withouttax,was$11.88,howmanydozenwerepurchased?

20. BestBuyShoeshadabacktoschoolspecial.Thetotalbillforfourpairsofshoescameto$69.24(beforetax.)Whatwastheaveragepriceforeachpairofshoes?

21. Ifyouonlypurchased1pairofshoesatBestBuyShoesinsteadofthefourdescribedinproblem20, howmuchwouldyouhavepaid,basedontheaverageprice?

Solveforx.Showyourwork.22. 6! = 72 23. 4! = 200 24. 3! = 50

25. 12! = 25.80 26.!! ! = 17.31 27. 4! = 69.24

28. 12! = 198 29. 1.98! = 11.88 30.!! ! = 2

31. Someoftheproblems22–30couldrepresenttheworkyoudidtoanswerquestions16–21.Writethenumberoftheequationnexttothestoryitrepresents.

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Period ______ Name: ______________________

Shh!! Please be Discreet (Discrete)!

A Solidify Understanding Task

1. TheLibraryofCongressinWashingtonD.C.isconsideredthelargestlibraryintheworld.Theyoftenreceiveboxesofbookstobeaddedtotheircollection.Sincebookscanbequiteheavy,theyaren’tshippedinbigboxes.If,onaverage,eachboxcontainsabout8books,howmanybooksarereceivedbythelibraryin6boxes,10boxes,ornboxes.

a. Useatable,agraph,andanequationtomodelthissituation.

b. Identifythedomainofthefunction

2. ManyofthebooksattheLibraryofCongressareelectronic.Ifabout13e-bookscanbedownloadedontothecomputereachhour,howmanye-bookscanbeaddedtothelibraryin3hours,5hours,ornhours(assumingthatthecomputermemoryisnotlimited)?

a. Useatable,agraphandanequationtomodelthissituation.

b. Identifythedomainofthefunction.

3. Woulditmakesenseinanyofthesesituationsfortheretobeatimewhen32.5bookshadbeenshipped,downloadedintothecomputerorplacedontheshelf?

AdaptedfromtheMathematicsVisionProject.LicensedundertheCreativeCommonsAttributionCCBY4.0.mathematicsvisionproject.org

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Ready

Statewhichsituationhasthegreatestrateofchange

1. Theamountofstretchinashortbungeecordstretches6incheswhenstretchedbyaa3poundweight.Aslinkystretches3feetwhenstretchedbya1poundweight.

2. Asunflowerthatgrows2incheseverydayoranamarylilisthatgrows18inchesinoneweek.3. Pumping25gallonsofgasintoatruckin2minutesorfillingabathtubwith40gallonsofwaterin

5minutes.4. Ridingabike10milesin1hourorjogging3milesin24minutes.

Set

Grapheachscenariothenidentifywhetherthefollowingitemsbestfitwithadiscreetorcontinuousmodel.Thendeterminewhattherateofchangeis.

1. Thefreewayconstructioncrewpours300ft.ofconcreteinaday.

2. Theaveragepersontakes10,000stepsinaday.

3. Foreveryhourthatpasses,theamountofareainfectedbythebacteriadoubles.

4. AttheheadwatersoftheMississippiRiverthewaterflowsatasurfacerateof1.2milesperhour.

AdaptedfromtheMathematicsVisionProject.LicensedundertheCreativeCommonsAttributionCCBY4.0.mathematicsvisionproject.org6

NOTES: Domain & Range - Discrete vs. ContinuousGoal: Students will conceptualize a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Warm-Up You and a friend are playing catch. You throw three different balls to your friend. You throw the first ball in an arc and your friend catches it. You throw the second ball in an arc, but this time the ball gets stuck in a tree. You throw the third ball directly at you friend, but it lands in front of your friend, and rolls the rest of the way on the ground. Match each graph with a situation from above.

Situation:___________________ Situation:__________________ Situation:__________________

Sketching Graphs for Situations

A continuous graph is _______________________________________________________________________________________________

________________________________________________________________________________________________________________________

A discrete graph is _______________________________________________________________________________________________

________________________________________________________________________________________________________________________

Example: Sketch a graph from the situation; tell whether the graph is continuous or discrete and determine the domain and range.

A student is taking a test. There are 10 problems on the test. For each problem the student answers correctly, the student receives 10 points.

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Example: A bathtub is being filled with water. After 10 minutes, there are 75 quarts of water in the tub. Then someone accidentally pulls the drain plug while the water is still running , and the tub begins to empty. The tub looses 5 quarts in 5 minutes, and then someone plugs the drain and the tub fills for 6 more minutes, gaining another 45 quarts of water. After a 15-minute bath, the person gets out and pulls the drain plug. It takes 11 minutes for the tub to drain.

You try: At the start of a snowstorm, it snowed two inches an hour for two hours, the slowed to one inch an hour for an additional hour before stopping. Three hours after it stopped, it began to melt at one-half inch an hour for two hours.

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HW: Domain & RangeComplete each sentence.

Find the domain and range for each graph. The first one is done for you.

3. 4. 5.

domain: domain:

domain: ________________________

_______________________ ________________________

range: range: range:

________________________ _______________________

________________________

For each situation, tell whether a graph would be a continuous graph or a discrete graph.

6. the number of cans collected for recycling _______________________

7. pouring a glass of milk ____________________________

8. the distance a car travels from a garage _________________________

9. the number of people in a restaurant ____________________________

1. The domain is the set of _________________________ numbers, or values of x.

2. The range is the set of ___________________________ numbers, or values of y.

0, 1, 2, 3, 4, 5

0, 1, 2, 3, 4, 5

Which graph represents the situation and has the correct domain and range.

10. Jason takes a shower, but the drain in the shower is not working properly.

a. b. c.

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NOTES - Intro to Linear Equations: The Parent Graph

Is there a pattern? Explain what it is.

Where is it touching the x-axis? The y-axis?

What if x = 7 ? What if x = 2.5 ?

What would be the domain and range?

Write a scenario that the graph would represent.

Parent Graph: _________________

x Y

-2 -2

-1 -1

0 0

1 1

2 2

WARMUP Graph the values from the table and use a straightedge to draw the line

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Examples

Given the parent graph, what would happen if…

1. The graph was shifted up 3 units?

Sketch the new line with respect to the parent graph.

Create a Table of Values for the New Line

𝑦 = 𝑥

2. The graph was shifted down 1 unit?



𝑦 = 𝑥

3. The graph was shifted down 1 unit?



𝑦 = 𝑥

x Y

x Y

x Y

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NOTES - LinearEquations:Identifying“m"

.

Whenlookingatthetwographswhatdoyounotice?

HowisthesecondGraphrelatedtotheparentgraph?

Howdoyourepresentthat?

WhatisSlope?

x Y

-2 -2

-1 -1

0 0

1 1

2 2

x Y

-2 -4

-1 -2

0 0

1 2

2 4

WARMUP Graph the following:

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Examples

Giventheparentgraph,whatwouldhappenif…

1. Theslopewas-1?Sketchthenewlinewithrespecttotheparentgraph.

CreateaTableofValuesfortheNewLine

𝑦 = 𝑥

2. Theslopewas½?Sketchthenewlinewithrespecttotheparentgraph.


𝑦 = 𝑥

3. Theslopewas-2?Sketchthenewlinewithrespecttotheparentgraph.


𝑦 = 𝑥

x Y

x Y

x Y

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NOTES - More About Rate of Change and SlopeLearning Targets: Students will use Rate of Change to solve problems

Students will be able to find the slope of a line.

1. Graph each relation. (– 𝟒, 𝟎), (−𝟐, 𝟏), (𝟐, 𝟎), (𝟐, 𝟑). Describe the pattern you see.

2. What is the Domain & Range.

a. Domain: ________________________________________

b. Range: __________________________________________

What is Slope?

WARM UP

DEFINITION CHARACTERISTICS

EXAMPLES/MODELS NON-EXAMPLES

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Finding Rate of Change from a Table.

𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 =𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑐𝑜𝑠𝑡

𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑔𝑎𝑚𝑒𝑠

Find the rate of change from the table.

SLOPE FROM A GRAPH

The slope of a nonvertical line is the ________________ of change in the __________ coordinates (RISE) to the

change in the ____________ coordinates (RUN) as you move from one point to another.

Rise= _________

Run= _________

Slope = ________

Number of Computer

Games

Total Cost ($)

2 78

4 256

6 434

Rise= _________

Run= _________

Slope = ________

Number of Floor Tiles

Area of Tiled Surface

3 48

6 96

9 144

Rise= _________

Run= _________

Slope = ________

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Finding Slope that passes through a pair of points

LINEAR EQUATIONS can have 4 different TYPES of SLOPE.

POSITIVE

(−𝟐, 𝟎)𝒂𝒏𝒅 (𝟏, 𝟓)

NEGATIVE

(−3,4)𝑎𝑛𝑑 (2, −3)

ZERO

(−3, −1)𝑎𝑛𝑑 (2, −1)

UNDEFINED

(−2,4) 𝑎𝑛𝑑 (−2, −3)

𝑆𝑙𝑜𝑝𝑒 = 𝑅𝑖𝑠𝑒

𝑅𝑢𝑛=

Δ𝑦

Δ𝑥=

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠

𝑺𝒍𝒐𝒑𝒆 = 𝒚𝟐 − 𝒚𝟏

𝒙𝟐 − 𝒙𝟏

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Name___________________________________

Date________________ Period____

HW

Finding Slope From a GraphFind the slope of each line.

1)

x

y2)

x

y

3)

x

y4)

x

y

5)

x

y6)

x

y

7)

x

y8)

x

y

9)

x

y10)

x

y

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NOTES – Characteristics of Linear Equations

Warmup – Identify the parts of the coordinate system below: 1)________________

2) ______________________

3) ____________

4)______________________ 5)_______________

NOTES – x-‐intercepts & y-‐intercepts

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Example 3: Graph using x and y intercepts.

4x − 2y = −6

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NAME: ____________________________________ DATE: ____________________________ PERIOD: _____________

HW Graphing Linear Equations & Intercepts

Find the x- and y-intercepts of each linear function.

1. 2. 3.

Graph each equation by making a table with 3 points.

4. y = 4 5. y = 3x 6. y = x + 4

Graph each equation by using the x- and y-intercepts.

7. x – y = 3 8. 10x + 5y =0 9. 4x + 2y = 6

Graph each equation.

10. !! 𝑥 – y = 2 11. 5x – 2y = 7 12. 1.5x + 3y = 9

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NAME: ____________________________________ DATE: ____________________________ PERIOD: ______________

Reminder --- Linear graphs can be translated on the coordinate plane. This means that the graph moves up, down, right, or left without changing its direction.

Translating the graphs up or down affects the y-coordinate for a given x value. Translating the graph right or left affects the x-coordinate for a given y-value.

Example: Translate the graph of y = 2x + 2, 3 units up.

y = 2x + 2

x y

–1 0

0 2

1 4

2 6

Add 3 to each y-value.

Translation

x y

–1 3

0 5

1 7

2 9

Graph the function and the translation on the same coordinate plane. You can use any of the methods we have learned to graph the given line. [HINT: each equation is in slope-intercept form already]

13. y = x + 4, 3 units down 14. y = 2x – 2, 2 units left

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NOTES: Slope-Intercept Form

Warmup:Graph using x

and y intercepts

What is the slope?

Slope-intercept Form

y = mx + b

slope y-intercept( 0, b )

Example 1: Graph

Example 2: Graph Example 3: Graph

Example 4: Graph Example 6: Graph

Example 5: Graph

Example 7: Graph

HW Write and graph 5 equations on your own paper. You MUST have at least two negative slopes :)

22

2

HW: Slope-Intercept Form

Write 5 equations and graph the corresponding lines on the grids provided. You MUST have at least two negative slopes :)

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NOTES: Functions Learning Targets: Students will determine whether a RELATION is a FUNCTION. Students will know how to use FUNCTION NOTATION to evaluate functions.

DEFINITION CHARACTERISTICS

EXAMPLES/MODELS NON-EXAMPLES

Determine whether each relation is a function? Explain.

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You can use the ___________________ _______________ _____________ to see if a graph

represents a function. If the vertical line intersects the graph MORE THAN ONCE, it is NOT a

function.

Equations that are functions can be written in a form called _____________ _____________. In the example above:

The INPUT is represented by which variable: _________. The OUTPUT is represented by which variable? ___________.

In FUNCTION NOTATION, 𝑓(𝑥) represents the range while 𝑥 represents the domain.

EQUATION

𝑦 = 2𝑥 + 4

FUNCTION NOTATION

𝑓 𝑥 = 2𝑥 + 4

𝑓(9) represents the OUTPUT value produced when the INPUT is 9.

Example 1: For 𝑓 𝑥 = −4𝑥 + 7, find each value. a. 𝑓 2 b. 𝑓 −2

Example 2: For ℎ 𝑡 = −16𝑡! + 68𝑡 + 2, find each value.

a. ℎ(4) b. ℎ 0

25

NAME DATE PERIOD

PDF Pass

HW Functions

Determine whether each relation is a function.

1. 2. 3.

4. x y

4 -5

-1 -10

0 -9

1 -7

9 1

5. x y

2 7

5 -3

3 5

-4 -2

5 2

6. x y

3 7

-1 1

1 0

3 5

7 3

7. {(2, 5), (4, -2), (3, 3), (5, 4), (-2, 5)} 8. {(6, -1), (-4, 2), (5, 2), (4, 6), (6, 5)}

9. y = 2x - 5 10. y = 11

11. 12. 13.

If f(x) = 3x + 2 and g(x) = x2 - x, find each value.

14. f(4) 15. f(8)

16. f(-2) 17. g(2)

18. g(-3) 19. g(-6)

20. f(2) + 1 21. f(1) - 1

22. g(2) - 2 23. g(-1) + 4

24. f(x + 1) 25. g(3b)

x

y

Ox

y

Ox

y

O

X Y

467

2-1

35

X Y

41

-2

520

-3

X Y

41

-3-5

-6-2

13

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26

PDF Pass

Determine whether each relation is a function.

1. 2. X Y

1 -5

-4 3

7 6

1 -2

3.

4. {(1, 4), (2, -2), (3, -6), (-6, 3), (-3, 6)} 5. {(6, -4), (2, -4), (-4, 2), (4, 6), (2, 6)}

6. x = -2 7. y = 2

If f(x) = 2x - 6 and g(x) = x - 2x2, find each value.

8. f(2) 9. f (- 1 − 2 ) 10. g(-1)

11. g (- 1 − 3 ) 12. f(7) - 9 13. g(-3) + 13

14. f(h + 9) 15. g(3y) 16. 2[g(b) + 1]

17. WAGES Martin earns $7.50 per hour proofreading ads at a local newspaper. His weeklywage w can be described by the equation w = 7.5h, where h is the number of hoursworked.

a. Write the equation in function notation.

b. Find f(15), f(20), and f(25).

18. ELECTRICITY The table shows the relationship between resistance R and current Iin a circuit.

Resistance (ohms) 120 80 48 6 4

Current (amperes) 0.1 0.15 0.25 2 3

a. Is the relationship a function? Explain.

b. If the relation can be represented by the equation IR = 12, rewrite the equation infunction notation so that the resistance R is a function of the current I.

c. What is the resistance in a circuit when the current is 0.5 ampere?

x

y

O

X Y

03

-2

-3-2

15

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27

Equation:______________________

Scenario: Sarah buys a plant that is 20 inches tall and grows two inches every week.

Equation:______________________

Scenario: Billy has 25 Skittles and eat two every hour.

Equation:______________________

Scenario: There are 42 passengers on the city bus. One person gets off the bus every three miles.

2.

1.

3.

4x4's ActivityFor each scenario, write an equation and then graph the line. Be sure to fill in the table of values.

28

1 3

1 5

3 7

4 8

6 10

Equation:______________________

Equation:______________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Equation:______________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

0 5

1 6.5

3 9.5

4 11

5 12.5

3 8

0 6

3 4

6 2

9 0

2 11

0 5

2 1

4 7

6 13

Equation:______________________

Scenario:_________________

_________________________

_________________________

For each table of values, write an equation and a scenario. Then graph the line.

29

Scenario: _____________________________________________________________

____________________________________

____________________________________

Equation:______________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Equation:______________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Equation:______________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Equation:______________________

For each line, write an equation and fill in the table of values. Then write a matching scenario.

30

Equation:

y = 2x + 4

Equation:

y = x + 3

Equation:

y = x 3

Equation:

y = 4x

Scenario: _____________________________________________________________

____________________________________

____________________________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

Scenario: _____________________________________________________________

____________________________________

____________________________________

For each equation, fill in the table of values and graph the line. Then write a matching scenario.

31

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32

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%0)(.)!.$ "!#Find the missing information and/or line for each problem.

33

NOTES: Finding x and y intercepts given 2 points

Find the x and y intercepts for the line going through the given points.

You Try1. ( -5, 1 ) & ( 3, -1 ) 2. ( 1, 3 ) & ( -3, -3 )

34

NOTES: Writing Equations From Graphs Find the slope of each line and then write the equation.

1) 2)

3) 4)

5) 6)

35

HW Writing Equations From Graphs Writing Equations HW #1 pp.318-319 #'s 8-11, 16-18, 24-26

36

NOTES: Writing Equations From Points Name___________________________________

Date________________ Period____REVIEW Slope m = y

2- y

1

x2

- x1

Slope- Intercept Form y = mx + b y - y1 = m(x - x

1)Point-Slope Form

Standard Form Ax + By = C

Write the equation of the line through the given points using TWO forms (slope-intercept, point-slope, and standard form).

1) through: (1, 5) and (2, -2) 2) through: (0, 2) and (-5, -1)

3) through: (2, -4) and (-2, 2) 4) through: (2, -4) and (2, -1)

5) through: (0, 5) and (5, 0) 6) through: (-3, -4) and (-1, -1)

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Eight segments are shown in the figure of a mountain range below. Find the slope of each segment and then match it with the slope of one of the functions to the right.

Record the letter with the exercise and in the correct space at the bottom of the page. The first one has been done for you. The mountain you will name is also known as “Denali” or “The Great One.”

Show your work on the next page provided.

1. slope =__ __; letter is K C. y =4

- 243

x

2. slope = ; letter is E. y =5

= - + 303

y x

3. slope = ; letter is I. y = –8x + 408

4. slope = ; letter is K. 5 15

= - 2 2

y x

5. slope = ; letter is L. y = –3x + 174

6. slope = ; letter is M. y = –x + 32

7. slope = ; letter is N. y = 4x – 204

8. slope = ; letter is Y. 3

- 482

y x

HW: Writing Equations Slippery Slope

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Making My Point

Activity

A Solidify Understanding Task

Zac and Sione were working on predicting the number of quilt

blocks in this pattern:

When they compared their results, they had an interesting discussion:

Zac: I got y = 6n + 1 because I noticed that 6 blocks were added each time so the pattern must

have started with 1 block at n = 0.

Sione: I got y = 6(n - 1) + 7 because I noticed that at n = 1 there were 7 blocks and at n = 2 there

were 13, so I used my table to see that I could get the number of blocks by taking one less than the

n, multiplying by 6 (because there are 6 new blocks in each figure) and then adding 7 because that's

how many blocks in the first figure. Here's my table:

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionpro ject.or g

WTl9mathematics vision project

1. What do you think about the strategies that Zac and Sione used? Are either of them correct?

Why or why not? Use as many representations as you can to support your answer.

41

The next problem Zac and Sione worked on was to write the equation of the line shown on the graph below.

When they were finished, here is the conversation they had about how they got their equations:

Sione: It was hard for me to tell where the graph crossed the y axis, so I found two points that I could read easily, (-9, 2) and (-15, 5). I figured out that the slope was - ½ and made a table and checked it against the graph. Here's my table:

X -15 -13 -11

J(x) 5 4


Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionpro ject.or g

3

-9 n

1 2 --(n + 9) + 22

WTl9mathematics vision project

I was surprised to notice that the pattern was to start with then, add 9, multiply by the slope and

then add 2.

I got the equation: f (x) = -½ (x + 9) + 2.

Zac: Hey-I think I did something similar, but I used the points, (7,-6) and (9,-7).

I ended up with the equation: f (x) = -½ (x - 9) - 7. One of us must be wrong because yours

says that you add 9 to then and mine says that you subtract 9. How can we both be right?

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2. What do you say? Can they both be right? Show some mathematical work to support your

thinking.

Zac: My equation made me wonder if there was something special about the point (9, -7) since it

seemed to appear in my equation f(x) = -½ (x - 9) - 7 when I looked at the number pattern.

Now I'm noticing something interesting-the same thing seems to happen with your equation,

f(x) =-½ex+ 9) + 2 and the point (-9, 2)

3. Describe the pattern that Zac is noticing.

4. Find another point on the line given above and write the equation that would come from

Zac's pattern.

5. What would the pattern look like with the point ( a, b) if you knew that the slope of the line

wasm?




WTl9mathematics vision project 43

6. Zac challenges you to use the pattern he noticed to write the equation of line that has aslope of 3 and contains the point (2,-1). What's your answer?

Show a way to check to see if your equation is correct.

7. Sione challenges you to use the pattern to write the equation of the line graphed below,using the point (S, 4).


8. Zac: ''I'll bet you can't use the pattern to write the equation of the line through the points(1,-3) and (3,-5). Try it!"






9. Sione: I wonder if we could use this pattern to graph lines, thinking of the starting point

and using the slope. Try it with the equation: f(x) = -2(x + 1) - 3.

Starting point: Slope:

Graph:

10. Zac wonders, ''What is it about lines that makes this work?" How would you answer Zac?

11. Could you use this pattern to write the equation of any linear function? Why or why not?





READY (Guided Practice)Topic: Writing equations of lines.

Write the equation of a line in slope-intercept form: y = mx + b, using the given information.

l. m=-7,b=4 2. m=3/8,b=-3 3. m = 16, b = -1/5

Write the equation of the line in point-slope form: y = m(x - x1)+ Y1, using the given information.

4. m=9,( 0.-7) 5. m = 2/3, (-6, 1)

7. (2,-5) (-3, 10) 8. (0, -9) (3, 0)



mathematicsvisionpro ject.or s

6. m = -5, (4, 11)

9. (-4, 8) (3, 1)

WT,.mathematics vision project

46

SET

Topic: Graphing linear and exponential functions

Make a graph of the function based on the following information. Add your axes. Choose an

appropriate scale and label your graph. Then write the equation of the function.

10. The beginning value is 5 and its value is 3

units smaller at each stage.

Equation:

12. The beginning value is 1 and its value is 10

times as big at each stage.

Equation:




11. The beginning value is 16 and its value is¼

smaller at each stage.

Equation:

13. The beginning value is -8 and its value is 2

units larger at each stage.

Equation:


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GO - HW

Topic: Equivalent equations

Prove that the two equations are equivalent by simplifying the equation on the right side of the

equal sign. The justification in the example is to help you understand the steps for simplifying.

You do NOT need to justify your steps.

Example:

2x - 4 = 8 + x - Sx + 6(x - 2) = 8 -4x + 6x -12 = -4+ 2x

2x - 4 = 2x -4

14. x - 5 = Sx - 7 + 2(3x + 1) - 10x

16. 14x + 2 = 2x - 3(-4x - 5) - 13

18. 4 = 7(2x + 1) -Sx -3(3x + 1)

Justification

Add x - Sx and distribute the 6 over (x - 2) Combine like terms.

Commutative property of addition

15. 6 - 13x = 24 - 10(2x + 8) + 62 + 7x

17. x+3= 6(x+3) - 5(x+3)

19. x= 12+8x-3(x+4)-4x

20. Write an expression that equals (x - 13). It must have at least two sets of parentheses and oneminus sign. Verify that it is equal to (x - 13).





48

WARMUP

Solve and graph the inequality

2x – 4 > 8

NOTES: Linear Inequalities

0

Graphing a linear inequality

• Identify the slope and y-intercept

• Determine solid (< or >) or dotted (< or >) line

• Choose a test point

• Shade above (> or >) or below (< or <) the line

Graph each inequality by choosing a test point.

49

NAME: ____________________________________ DATE: ____________________________ PERIOD:______________

HW Graphing Inequalities in Two Variables

Match each inequality to the graph of its solution.

1. y – 2x < 2 a. b.

2. y ≤ –3x

3. 2y – x ≥ 4

4. x + y > 1

c. d.

Graph each inequality.

5. y < –1 6. y ≥ x – 5 7. y > 3x

8. y ≤ 2x + 4 9. y + x > 3 10. y – x ≥ 1

50

NAME: ____________________________________ DATE: ____________________________ PERIOD: ______________

Determine which ordered pairs are part of the solution set for each inequality.

11. 3x + y ≥ 6 {(4, 3), (–2, 4), (–5, –3), (3, –3)}

12. y ≥ x + 3 {(6, 3), (–3, 2), (3, –2), (4, 3)}

13. 3x – 2y < 5 {(4, –4), (3, 5), (5, 2), (–3, 4)}

Graph each inequality.

14. 2y – x < –4 15. 2x – 2y ≥ 8 16. 3y > 2x – 3

51

WARMUP

GraphthefollowingInequality:

2𝑥 + 3𝑦 > 6

y-int:____________

x-int:____________

Slope:___________Whichofthefollowingwouldbesolutionstotheinequality?Whyorwhynot?

(0,0) (-3,4) (0,3) (-4,2)

Review of Function Notation:

𝑓 𝑥 = 𝑥! + 3

Find eachofthefollowing:

𝑓 −2 = 𝑓 2 =

NOTES: Explicit and Recursive Rules

52

Recursive/Explicit

RecursiveFunction: ExplicitFunction:

Writearecursiveformulafor: Writeanexplicitformulafor:70,77,84,91,… 70,77,84,91,…

Writeanexplicitformulafromarecursiveformula:Thefirsttermis19,so𝐴 1 = 19

𝐴 𝑛 = 𝐴 𝑛 − 1 + 12

YourTurn:𝐴 1 = 21;𝐴 𝑛 = 𝐴 𝑛 − 1 + 2

53

Name___________________________________

Date________________ Period____

Unit 2 Review

Find the slope of each line.

1)

x

y2)

x

y

Find the slope of the line through each pair of points.

3) (19 , 8 ), (11 , - 16 ) 4) (- 4 , - 1 ), (- 16 , 3 )

Find the slope and y-intercept of each line.

5) y = - 4 x + 5 6) y = x - 2

Sketch the graph of each line.

7) x-intercept = - 1 , y-intercept = 5

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

8) x-intercept = 1 , y-intercept = 4

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

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9) y = -4

3x + 2

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

10) y = - x - 5

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

11) 2 x - 5 y = 15

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

12) x + 2 y = - 4

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

Write the slope-intercept form of the equation of each line.

13)

x

y

- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

- 4

- 2

2

4

14)

x

y

- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

- 4

- 2

2

4

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Write the slope-intercept form of the equation of each line given the slope and y-intercept.

15) Slope = 1 , y-intercept = - 3 16) Slope = 2 , y-intercept = 5

Write the slope-intercept form of the equation of the line through the given point with the givenslope.

17) through: (3 , 4 ), slope = 8

318) through: (4 , - 1 ), slope =

1

2

Sketch the graph of each linear inequality.

19) y £ 3 x - 4

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

20) y < - x + 2

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

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21) x + 5 y £ 10

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

22) x - y > - 2

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

23) Determine the domain and range of thefollowing relation {(3, 5), (–4, 6), (3, 8), (2, 4), (1, 3)}.

24) Express each relation as a graph. Thendetermine the domain and range.

{(–1, –1), (1, 1), (2, 1), (3, 2)}

x

y

- 6 - 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6

- 6

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

Domain:__________________

Range:___________________

Domain:__________________

Range:___________________

57

Algebra 1 Unit 2

Documents

Transcript of Algebra 1 Unit 2