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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
3
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.
4
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
5
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.
7
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.
8
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.
9
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
10
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?
Sketch the new line with respect to the parent graph.
Create a Table of Values for the New Line
𝑦 = 𝑥
3. The graph was shifted down 1 unit?
Sketch the new line with respect to the parent graph.
Create a Table of Values for the New Line
𝑦 = 𝑥
x Y
x Y
x Y
11
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:
12
Examples
Giventheparentgraph,whatwouldhappenif…
1. Theslopewas-1?Sketchthenewlinewithrespecttotheparentgraph.
CreateaTableofValuesfortheNewLine
𝑦 = 𝑥
2. Theslopewas½?Sketchthenewlinewithrespecttotheparentgraph.
CreateaTableofValuesfortheNewLine
𝑦 = 𝑥
3. Theslopewas-2?Sketchthenewlinewithrespecttotheparentgraph.
CreateaTableofValuesfortheNewLine
𝑦 = 𝑥
x Y
x Y
x Y
13
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
14
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 = ________
15
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
16
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
17
NOTES – Characteristics of Linear Equations
Warmup – Identify the parts of the coordinate system below: 1)________________
2) ______________________
3) ____________
4)______________________ 5)_______________
NOTES – x-‐intercepts & y-‐intercepts
18
Example 3: Graph using x and y intercepts.
4x − 2y = −6
19
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
20
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
21
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 :)
23
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.
24
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
042_054_ALG1_A_CRM_C01_CR_660498.indd 46042_054_ALG1_A_CRM_C01_CR_660498.indd 46 12/21/10 5:21 PM12/21/10 5:21 PM
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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7++’++#’(0Fill in the missing information and/or line for each problem.
32
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C";</52651/= D E FG";</52651/= D E F
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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
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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:
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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.
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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
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3
-9 n
1 2 --(n + 9) + 22
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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?
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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).
Show a way to check to see if your equation is correct.
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!"
Show a way to check to see if your equation is correct.
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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?
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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)
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6. m = -5, (4, 11)
9. (-4, 8) (3, 1)
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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:
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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).
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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.
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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
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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
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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
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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
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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:___________________
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