Continuity of Learning Assignments Essential …...Continuity of Learning Assignments Grade: _9-12___ Subject: _Math_____ Days 1-2 Days 3-4 Days 5-6 Days 7-8 Days 9-10 Essential Activity

Continuity of Learning Assignments

Grade: _9-12___ Subject: _Math________

Days 1-2 Days 3-4 Days 5-6 Days 7-8 Days 9-10 Essential Activity Essential Activity Essential Activity Essential Activity Essential Activity

Digital Learning Algebra Nation Students have been assigned the following lessons

x Section 3: Introduction to Functions

x Intro, Topic 1

Algebra Nation on Paper: Complete Pages 1-4



x Topics 2-3 Algebra Nation on Paper: Complete Pages 5-9










Extension Extension Extension Extension Extension Digital Learning:

x Eureka Math x Khan Academy

Digital Learning:


Digital Learning:


Digital Learning:


Digital Learning:


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Name _____________________________________________ Date _________________ Introduction to Functions Input and Output Values Independent Practice 1. A giraffe’s hunger level depends on the size of its last meal.

Part A: What is the independent variable?

Part B: What is the dependent variable? 2. You earn $20 per hour doing landscaping work. Your total earnings depend on the

amount of hours you spend landscaping.

Part A: What is the independent variable?

Part B: What is the dependent variable?

Part C: Write a function to represent the situation. 3. Mr. Khans is buying staplers for his office. Each stapler costs $16.99.

Part A: What does his final total cost depend upon? Part B: In this scenario, what is the input? What is the output? Part C: Write a function to represent the situation. Part D: If Mr. Khans buys 15 staplers, it would cost him $254.85. How would you write this using function notation?

4. Cristiano Ronaldo has a messy house. The function that represents the total cost

Cristiano spends having his house cleaned is + , = 125,, where , represents the number of visits. Part A: How much does Cristiano spend per visit? Part B: Define the input and output in the given scenario.

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5. Consider the following incomplete mapping diagrams.

Part A: Complete Diagram A so that it is a function.

Part B: Is it possible to complete Diagram B so that it is NOT a function. If so, complete the diagram to show a relation, but not a function. If not, justify your reasoning.

Part C: Is it possible to complete the mapping diagram for Diagram C so it represents a function? If so, complete the diagram to show a function. If not, justify your reasoning.

6. The cost to manufacture . chairs can be represented by the function + . = 36.. Circle the pair of numbers (one in each box) that correctly completes the statement about the function.

If +(63) = 2268, then chairs cost $ 7. Which of the following relations are not functions? Select all that apply.

o 1, 3 , 3, 7 , 5, 11 , 7, 15 , (9, 19) o 1, 3 , 1, 7 , 5, 11 , 5, 15 , (9, 19) o −2, 4 , −1, 1 , 0, 0 , 1, 1 , (2, 4) o 2, 4 , 1, 1 , 0, 0 , 1, −1 , 2, −4 o 6, 3 , 4, 1 , 2, 1 , 0, −1 , (−2,−3) o 1, 3 , 3, 7 , 3, 11 , 7, 15 , (9, 19) o 1, 3 , 3, 7 , 5, 11 , 9, 15 , (9, 19)

32-1

555

555

32-1

555

32-1

Diagram A Diagram B Diagram C

0663378

6.189.378.2,268.

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Name _______________________________________________ Date _______________ Introduction to Functions Representing, Naming, and Evaluating Functions Independent Practice 1. A ball is thrown into the air with an initial velocity of 22 meters per second. The

quadratic function ℎ($) = −4.9$+ + 22$ + 5.5 represents the height of the ball above the ground, in meters, with respect to time $, in seconds.

Part A: Determine ℎ(3) and explain what it represents.

The graph below represents the height of the ball with respect to time.

Part B: What is a reasonable domain for the function?

Part C: What is a reasonable range for the function?

Height of the Ball Over Time

Time, in seconds

Heig

ht, i

n m

eter

s

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2. On the moon, the time, in seconds, it takes for an object to fall a distance, /, in feet, is given by the function 0 / = 1.11 /.

Part A: Determine 0(2) and explain what it represents.

Part B: The Imbrium Basin is the largest basin on the moon. A reasonable domain for the height above the lowest point in the basin is given by / 0 ≤ / ≤ 3805774 . What does this tell you about the basin?

Part C: How long would it take a rock to drop from the rim to the bottom of the basin?

3. The function that represents the amount of caffeine, in milligrams, remaining in a body after drinking the two Mountain Dew sodas is given by 0 $ = 110 0.8855 6 where $ is time in hours. How much caffeine is left in a person’s body 18hours after drinking two Mountain Dew sodas?

4. The function 0 $ = 98.6 − 9 , represents a person’s variance from normal body temperature, where 9 represents a person’s current body temperature in degrees Fahrenheit. Medical professionals say healthy individuals should have a variance of no more than 0.5℉. Part A: What can be said of an individual with a body temperature of 99.4℉?

Justify your answer.

Part B: What can be said of an individual with a body temperature of 98.4℉? Justify your answer.

Part C: What is a reasonable domain that would indicate the individual is not healthy?

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Name _____________________________________________ Date_______________ Introduction to Functions Adding and Subtracting Functions Independent Practice

1. Let ! " = 7"% − 5" + 3 and * " = 2"% + 4" − 6.

Part A: Find . / + 0(/) Part B: Find . / − 0(/) Part C: Find 0 / − . /

2. The perimeter of the triangle below is 3/ + 56. Find the measure of the missing side.

3. The perimeter of a trapezoid is 578– :. Three sides have the following

lengths: 78, ;8 + <, and <:8– =. What is the length of the fourth side?

4. Jocelyn and Lorlesha are comparing the size of their villages in the Clash of

Clans app. The area of Jocelyn’s village is represented by the polynomial, >?> + <@? + <>. The area of Lorlesha’s village is represented by the polynomial, 5?> + 3? − ;, where ? represents the width, in meters of their Town Hall. Part A: Find the expression that represents the additional area of Jocelyn’s village. Part B: Find the expressions that represents the combined total area of their villages.

! − $ ! + $

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Name ________________________________________________ Date _______________ Introduction to Functions Multiplying Functions Independent Practice 1. Evaluate !(#) ∙ ℎ(#) by modeling or by using the distributive property.

! # = (# − 2) and ℎ # = (#* + 4#- − 2)

2. Evaluate .(/) ∙ !(/) by modeling or by using the distributive property.

. / = (−3/ + 2) and ! / = (2/- − 5/ − 1)

3. Jamie used the distributive property to find the product of 3(4) and ℎ(4). His work was marked incorrect. Identify Jamie’s mistake. What advice would you give Jamie to avoid this mistake in the future.

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4. The figure below shows the penalty box and the goal box of a soccer field. The penalty box is the larger rectangle.

Part A: Find the area of the penalty box.

Part B: Find the area of the goal box.

Part C: Find the area of the penalty box not covered by the goal box.

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Name _______________________________________________ Date ______________ Introduction to Functions Closure Property Independent Practice 1. For the following exercises determine if the closure property applies to the following

statements by circling ‘True’ or ‘False’. Then provide an example of each statement.

Statement A Answer Choice Polynomials are closed under addition. True False Example:

Statement B Answer Choice Polynomials are closed under subtraction. True False Example:

Statement C Answer Choice Polynomials are closed under multiplication. True False Example:

Statement D Answer Choice Polynomials are closed under division. True False Example:

2. Check the boxes for the following sets that are closed under the given operations.

Set + - x ÷

{...−7,−6,−5,−4,−3} � � � �

{0, 1, 4, 9, 16, 25...} � � � �

{...−012 ,−

32,−

42,−

52,−

22...} � � � �

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3. Consider the following polynomials.

672 + 367 + 862 −5672

Use the two polynomials to illustrate the following: a. Polynomials are closed under addition. b. Polynomials are closed under subtraction c. Polynomials are closed under multiplication. d. Polynomials are not closed under division.

4. Ms. Adams claims that the closure properties for polynomials are closed when

dividing polynomials. Mr. Aroxa claims that the closure properties for polynomials are not closed when dividing polynomials. Who is correct? Explain your answer.

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Name _______________________________________________ Date ______________ Introduction to Functions Real-World Combinations and Compositions of Functions Independent Practice 1. The student government association is selling roses for Valentine’s Day to raise

money for their trip to the state convention. The cost of each rose is $". $% and the florist charges a delivery fee of $&$. The class plans to sell the roses for $$. %% each.

a. Define the variable.

b. Write a cost function.

c. Write a revenue function.

d. Write a profit function.

2. A local civic group is selling t-shirts to raise funds for Relay to Life. They plan to sell

&$%%t-shirts for $"%. They consider raising the t-shirt price in order to earn more profit. For each $" increase, they will sell "%% fewer t-shirts. Let ( represent the number of $" increases.

a. Write a function, )((), to represent the cost of one t-shirt based on the number

of increases.

b. Write a function, ,((), to represent the number of t-shirts sold based on the number of increases.

c. Write a revenue function, -((), for the t-shirt sell that could be used to maximize

revenue.

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3. Anna gets paid $.. /$/hour working as a barista at Coffee Break. Her boss pays her $0. %%/hour for creating the weekly advertisement signs. She works a total of &$ hours each week.

a. Let ( represent the hours that Anna works each week as a barista. Write a

function 1(() to represent the amount of money that Anna earns working as a barista.

b. Write a function, 2(() to represent the hours Anna works creating the signs.

c. Let 3 represent the number of hours that Anna works creating the signs. Create a function 4(3) to represent the money Anna earns creating the signs.

d. Find 4 2 ( . What does this composite function represent?

e. What functions could be combined to represent Anna’s total earnings? Combine the functions to write an expression that can be used to represent Anna’s total earnings, where ( represents the number of hours she works as a barista.

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4. Consider the following standards. MAFS.912.F-BF.1.1b: Combine standard function types using arithmetic operation. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

MAFS.912.F-BF.1.1c: Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. a. Describe the difference in combining functions and composing functions.

b. Give a real world example of combining functions.

c. Give a real world example of composing functions.

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Name Date Introduction to Functions Key Features of Graphs of Functions – Part 1 Independent Practice 1. The following statement is false. Highlight the two words that should be

interchanged to make it a true statement.

In a function, every output value corresponds to exactly one input value.

2. The following graph fails the vertical line test and is not a function.

a. Explain how the vertical line test shows that this relation is NOT a function.

b. Name two points on the graph that show that this relation is NOT a function. 3. Sketch the graph of a relation that is a function.

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4. Sketch the graph of a relation that is NOT a function.

5. Consider the following scenarios. Determine if each one represents a function or

not. Explain your answer. a. A golf ball is hit down a fairway. The golfer relates the time passed to the

height of the ball. b. A trainer takes a survey of all the athletes in a school about their height,

rounded to the nearest inch, and their grade level. The trainer relates their grade levels to their heights.

6. Use the word bank to complete the sentences below. ! −coordinate # −coordinate ! −intercept # −intercept solution

a. The ____________________ of a graph is the location where the graph crosses the

! −axis. b. The ____________________ of a graph is the location where the graph crosses the

# −axis. c. The ____________________ of the # −intercept is always zero.

d. The ____________________ of the ! −intercept is always zero.

e. The ! −intercept is the ______________ to a function or group.

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7. The graph below represents a rock climber’s height as she ascends a hill.

a. The above graph is (circle one) linear/nonlinear.

b. Is the above graph a function? Explain.

c. What is the # −intercept and what does the # −intercept represent?

d. Why would there not be an ! −intercept for this situation?

Time (s)

Elev

atio

n (ft

)

Rock Climbing

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8. The graph below represents the path of a golf ball.

a. The above graph is (circle one) linear/nonlinear.

b. Is the above graph a function? Explain.

c. What is the # −intercept and what does the # −intercept represent?

d. What is the solution to this graph and what does it represent in this situation?

Distance (ft)

Elev

atio

n (ft

)

The path of a golf ball

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Name ____________________________________________ Date_______________ Introduction to Functions Key Features of Graphs of Functions – Part 2 Independent Practice 1. Consider the following graph of an absolute value function.

a. Define the domain. b. Define the range. c. Where is the graph increasing? d. Where is the graph decreasing? e. Identify any relative maximums. f. Identify any relative minimums.

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2. Consider the following graph of a quadratic function.


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3. Consider the following graph of an exponential function.


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Name ____________________________________________ Date_______________ Introduction to Functions Average Rate of Change Over an Interval Independent Practice 1. Suppose that the cost of producing ! tablets is defined by

" ! = $%% + '%! + %. $!$, where ! represents the number of tablets produced. The graph below represents the function.

Part A: Complete the boxes to label the graph. Part B: Find the average rate of change over the interval [*, ,]. Part C: Find the average rate of change over the interval [,, .]

Part C: Compare the average rate of change over the interval *, , with his average rate of change over the interval [,, .]. What does this represent in real life?

2. Consider the table for the exponential function, / 0 = 22, shown below.

Part A: Which of the following intervals would you expect to have the greatest rate of change? Explain your reasoning. Part B: Find the rate of change of the three intervals to determine if your hypothesis in Part A was correct.

Point ! 3 !

4 0 1

7 2 4

9 4 16

; 6 64

!"

#

(15,395)(20,480)

(25,575)

o [4, 7]o [7, 9]o [9, ;]

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4. Consider the following graph of a quadratic function.

Part A: How can you quickly determine the intervals with the greatest rate of change by making observations from the graph?

Part B: Which intervals have the greatest rate of change?

Part C: Which intervals have the least rate of change?

Part D: Name two intervals have equal rates of change?

4. Bradford hopped into his Ford Mustang GT to rush to Diana and tell her how fantastic his students did on the End of Course exam. His Mustang can go from 0 to 60 mph in a brief 4.7seconds. Bradford accelerated to top speed quickly and then maintained a speed of 60 mph until he arrived at Diane’s office.

Part A: Sketch the graph of the situation on the coordinate plane.

Part B: Over which interval would you have the greatest rate of change? Justify your answer.

Part C: What part of the situation represents the time when the graph has no rate of change?

!

"

#

$

%

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Name ______________________________________________ Date _______________ Introduction to Functions Transformations of Functions Independent Practice 1. Label the following as transformations on the independent variable or the

dependent variable and describe the transformation. Description

a. ! " + 3

b. ! " − 3

c. !(" + 3)

d. !(" − 3)

2. The following table represents the function ℎ " . Complete the table for *("), given

* " = ,- ℎ " .

. /(.) . 0(.)

−4 256

−2 16

0 0

3 81

6 1296

o Independent Variable o Dependent Variable




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3. The following graph represents the function ! " . Sketch and label the following functions on the same coordinate plane. a. ! " + 2 b. !(" − 5) c. ! " + 2 d. ! " − 5

4. The following table represents the function ℎ " . Complete the table for *("), given * " = ℎ ,

9 " . The first two have been done for you!

. /(.) . / :;. 0(.)

−2 3.25 −8 ℎ 14 ∙ −8 = ℎ(−2) 3.25

−1 3.5 −4 ℎ 14 ∙ −4 = ℎ(−1) 3.5

0 4

1 5

2 7

3 11

4 19

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5. Consider the following table of values.

. >(.) 0(.) / . ?(.)

−5 25 23 −25 32

−3 9 7 −9 16

0 0 −2 0 7

3 9 7 −9 16

5 25 23 −25 32

a. Write *(") as a transformation of !(").

b. Write ℎ(") as a transformation of !(").

c. Write @(") as a transformation of !(").

6. Consider the following table of values.

. >(.) . ?(.)

−5 25 −7 25

−3 9 −5 9

0 0 −2 0

3 9 1 9

5 25 3 25

a. Write @(") as a transformation of !(").

b. Write !(") as a transformation of@(").

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7. Consider the following graph.

a. Write ℎ(") as a transformation of *(").

b. Write *(") as a transformation of !(").

c. Write ℎ(") as a transformation of ! " .

8. Consider the following Algebra 1 standard that deals with transformations.

MAFS.912.F-BF.2.3: Identify the effect on the graph of replacing !(") by !(") + C, C!("), !(C"), and !(" + C) for specific values of C(both positive and negative); find the value of C given the graphs. a. Circle the expressions that describe a transformation on the independent

variable.

b. Underline the expressions that describe a transformation on the dependent variable.

!(")

*(")

ℎ(")

Section 3: Introduction to Functions53

Section 3: Introduction to Functions Section 3 – Topic 1

Input and Output Values A function is a relationship between input and output.

�� Domain is the set of values of , used for the ___________ of the function.

�� Range is the set of values of - calculated from the

domain for the __________ of the function. In a function, every , corresponds to only one -.

�� - can also be written as ! , � Consider the following function.

For every , there is a unique �- .

input output domain range

��

��

We also refer to the variables as independent and dependent. The dependent variable ______________ ____ the independent variable. Refer to the mapping diagram on the previous page. Which variable is independent? Which variable is dependent? Consider a square whose perimeter depends on the length of its sides. What is the independent variable? What is the dependent variable? How can you represent this situation using function notation?

�

We can choose any letter to represent a function, such as !M,N or "M,N, where , is the input value. By using different letters, we show that we are talking about different functions. �


0

Let’s Practice! 1.� You earn �� per hour babysitting. Your total earnings

depend on the number of hours you spend babysitting.

a.� What is the independent variable?

b.� What is the dependent variable?

c.� How would you represent this situation using function notation?

2.� The table below represents a relation.

2 3 a.� Is the relation also a function? Justify your answer.

b.� If the relation is not a function, what number could be changed to make it a function?

D� �

��

�

D� �

Try It! 3.� Mrs. Krabappel is buying composition books for her

classroom. Each composition book costs ��.

a.� What does her total cost depend upon?

b.� What are the input and output?

c.� Write a function to describe the situation.

d.� If Mrs. Krabappel buys � composition books, they will cost her ��. Write this function using function notation.


4.� Consider the following incomplete mapping diagrams.

a.� Complete Diagram A so that it is a function.

b.� Complete Diagram B so that it is NOT a function.

c.� Is it possible to complete the mapping diagram for Diagram C so it represents a function? If so, complete the diagram to show a function. If not, justify your reasoning.

��

��

Diagram A

��

��

Diagram B

��

Diagram C

��

BEAT THE TEST! 1.� Isaac Messi is disorganized. To encourage Isaac to be

more organized, his father promised to give him three dollars for every day that his room is clean and his schoolwork is organized.

Part A: Define the input and output for the given

scenario.

Input:

Output:

Part B: Write a function to represent this situation.


2.� The cost to manufacture , pairs of shoes can be represented by the function � , G ,. Complete the statement about the function.

If �M N G ��, then pairs of shoes cost

3.� Which of the following relations is not a function? �� O �� P �� O �� D�� P �� O � � � �� D� � �� P �� OMD, DN, M� �N, M�, DNP

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Want some help? You can always ask questions on the Algebra Wall and receive help from other students, teachers, and Study Experts. You can also help others on the Algebra Wall and earn Karma Points for doing so. Go to AlgebraNation.com to learn more and get started!

��

� ��

Section 3 – Topic 2 Representing, Naming, and Evaluating Functions

A ball is thrown into the air with an initial velocity of �� meters per second. The quadratic function #M*N G D��*8 C ��* C represents the height of the ball above the ground, in meters, with respect to time *, in seconds. Determine #MN and explain what it represents. Is D a reasonable input for the function? The graph below represents the height of the ball with respect to time.

� What is a reasonable domain for the function? What is a reasonable range for the function?

Time (in seconds)

Heig

ht (i

n m

eter

s)

Height of the Ball Over Time


0

Let’s Practice! 1.� On the moon, the time, in seconds, it takes for an object

to fall a distance, , in feet, is given by the function ! G �� .

a.� Determine !M�N and explain what it represents.

b.� The South Pole-Aitken basin on the moon is �� feet deep. Determine a reasonable domain for a rock dropped from the rim of the basin.

2.� Floyd drinks two Mountain Dew sodas in the morning. The

function that represents the amount of caffeine, in milligrams, remaining in his body after drinking the sodas is given by ! * G �� @ where * is time in hours. Floyd says that in two days the caffeine is completely out of his system. Do you agree? Justify your answer.

Try It! 3.� Medical professionals say that �� is the normal body

temperature of an average person. Healthy individuals’ temperatures should not vary more than �� from that temperature.

a.� Write an absolute value function !M*N to describe an

individual’s variance from normal body temperature, where * is the individual’s current temperature.

b.� Determine !M��N and describe what that tells you about the individual.

c.� What is a reasonable domain for a healthy individual?


0

BEAT THE TEST! 1.� The length of a shipping box is two inches longer than the

width and four times the height.

Part A: Write a function �M+N that models the volume of the box, where + is the width, in inches.

Part B: Evaluate �M��N. Describe what this tells you about

the box.

��

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Section 3 – Topic 3 Adding and Subtracting Functions

Let # , G ,8 C , D � and " , G D,8 C �, C �. Find # , C "M,N. Find # , D " , .


0

Let’s Practice! 1.� Consider the following functions.

! , G ,8 C , C " , G �,8 C , D �

# , G �M,8 D �N

a.� Find ! , D " , .

b.� Find " , D # , �

Try It! 2.� Recall the functions we used earlier.

! , G ,8 C , C

" , G �,8 C , D � # , G �M,8 D �N

a.� Let &M,N be ! , C " , . Find &M,N.

b.� Find # , D& , .


BEAT THE TEST! 1.� Consider the functions below.

! , G ,8 C , D � " , G �,8 C �, D �

Which of the following is the resulting polynomial when ! , is subtracted from "M,N? �� D,8 D , D �� D,8 C �, D �� ,8 C , C �� ,8 C �, D �

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Section 3 – Topic 4 Multiplying Functions

Use the distributive property and modeling to perform the following function operations. Let ! , G ,8 C �, C and " , G , C . Find !M,N � "M,N.


Let & - G -; D -8 C � and ' - G -8 D . Find &M-N � 'M-N.

Let’s Practice! 1.� Let # , G , D � and " , G ,9 C ,8 D �.

Find #M,N � "M,N.


Try It! 2.� The envelope below has a mailing label. �

a.� Let � , G � , � � , D�M,N � �M,N. Find �M,N.

�M,N G , C �

MR. AL GEBRA 123 INFINITY WAY POLYNOMIAL, XY 11235

�M,N G , C � �M,N G , C �

�M,N

G,C

b.� What does the function �M,N represent in this problem?


0

BEAT THE TEST! 1.� The length of the sides of a square are ) inches long.

A rectangle is six inches shorter and eight inches wider than the square. Part A:� Express both the length and the width of the

rectangle as a function of a side of the square. Part B:� Write a function to represent the area of the

rectangle in terms of the sides of the square.

2.� Felicia needs to find the area of a rectangular field in her backyard. The length is represented by the function �� , G �,: D ,8 C and the width is represented by the function�� , G , C �. Which of the following statements is correct about the area, � , ��of the rectangular field in Felicia’s backyard? Select all that apply.� �� , G Q� , C� , R �� The resulting expression for �M,N is a fifth-degree

polynomial. �� The resulting expression for �M,N is a polynomial with a

leading coefficient of �. �� The resulting expression for �M,N is a binomial with a

constant of . �� , G >MAN

?MAN

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0

Section 3 – Topic 5 Closure Property

When we add two integers, what type of number is the sum? When we multiply two irrational numbers, what type of numbers could the resulting product be? A set is ___________ for a specific operation if and only if the operation on two elements of the set always produces an element of the same set. Are integers closed under addition? Justify your answer. Are irrational numbers closed under multiplication? Justify your answer.

Let’s apply the closure property to polynomials. Are the following statements true or false? If false, give a counterexample. Polynomials are closed under addition. Polynomials are closed under subtraction. Polynomials are closed under multiplication.


0

Let’s Practice! 1.� Check the boxes for the following sets that are closed

under the given operations.

Set D E F

O�� P ��

O� �E��E�E�E�P ��

O� �E�E�E�� P ��

Orational numbersP ��

OpolynomialsP ��

Try It! 2.� Ms. Sanabria claims that the closure properties for

polynomials are analogous to the closure properties for integers. Mr. Roberts claims that the closure properties for polynomials are analogous to the closure properties for whole numbers. Who is correct? Explain your answer.


0

BEAT THE TEST! 1.� Choose from the following words and expressions to

complete the statement below.

�-=7 C �,8 C �-8

The product of �,: D ,8 C and _______________________

illustrates the closure property because the

_______________ of the product are ____________________ ,

and the product is a polynomial.

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,; C M-N=8 D M�-N8 C �, C -9

integers variables whole numbers coefficients rational exponents

numbers

Section 3 – Topic 6 Key Features of Graphs of Functions – Part 1

Let’s review the definition of a function. Every input value M.N corresponds to ___________ _______ output value M/N� Consider the following graph.

How can a vertical line help us quickly determine if a graph represents a function? We call this the vertical line test. Use the vertical line test to determine if the graph above represents a function.


Important facts:

�� Graphs of lines are not always functions. Can you describe a graph of a line that is not a function?

�� Functions are not always linear.

Sketch a graph of a function that is not linear.

Let’s Practice! 1.� Use the vertical line test to determine if the following

graphs are functions.

Try It! 2.� Which of the following graphs represent functions? Select

all that apply.

��

��

/� /� /� /�

/� /� /� /�

.� .� .� .�

.� .� .� .�


0

It’s important to understand key features of graphs.

�� An 2-intercept of a graph is the location where the graph crosses the _____________.

�� The --coordinate of the ,-intercept is always _______. �� The 3-intercept of a graph is the location where the

graph crosses the _____________. �� The ,-coordinate of the --intercept is always

__________. �� The ,-intercept is the _____________ to ! , G �.

All of these features are very helpful in understanding real-world context.

3.� Consider the following scenarios. Determine if each one represents a function or not. a.� An analyst takes a survey of people about their

heights (in inches) and their ages. She then relates their heights to their ages (in years).

b.� A geometry student is dilating a circle and analyzes the area of the circle as it relates to the radius.

c.� A teacher has a roster of students and relates the students’ letter grades to the percent of points earned.

d.� A boy throws a tennis ball in the air and lets it fall to the ground. The boy relates the time passed to the height of the ball.


0

Let’s Practice! 4.� Consider the following graph that represents the height, in

feet, of a water balloon dropped from a 2nd story window after a given number of seconds.

a.� What is the ,-intercept? b.� What is the --intercept? c.� Label the intercepts on the graph.

Try It! 5.� Refer to the previous problem for the following questions.

a.� What does the --intercept represent in this real-world context?

b.� What does the ,-intercept represent in this real-world

context? c.� What is the solution to this situation?

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Let’s Practice! 1.� Use the following graph of an absolute value function to

answer the questions below.

a.� Define the domain using set notation.

b.� Define the range using interval notation.

c.� Where is the graph increasing?

d.� Where is the graph positive?

e.� Identify any relative maximums.

f.� Identify any relative minimums.

Section 3 – Topic 7 Key Features of Graphs of Functions – Part 2

Let’s discuss other key features of graphs of functions.

�� Domain: the input or the �� values. �� Range: the ____________________ or the /-values.

�� The domain and range of a function can be expressed

in set notation or interval notation. �� Increasing intervals: as the .-values _________________,

the /-values _________________.

�� Decreasing intervals: as the .-values _________________, the /-values _________________.

�� Relative maximum: the point on a graph where the

interval changes from __________________ to __________________.

�� Relative minimum: the point on a graph where the

interval changes from __________________ to __________________.

�� Positive intervals: intervals of a function #M.N over which

_________________.

�� Negative intervals: intervals of a function #M.N over which _________________.

�


0

Try It! 2.� Use the graph of the following quadratic function to

answer the questions below. Use set notation to express all answers.

a.� Define the domain.

b.� Define the range.

c.� Where is the graph decreasing?

d.� Where is the graph negative?

e.� Identify any relative maximums. f.� Identify any relative minimums.

3.� Describe everything you know about the key features of the following graph of an exponential function. Use interval notation to express all answers.


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Section 3 – Topic 8 Average Rate of Change Over an Interval

Consider the following graph of the square root function #M.N G �. .

Draw a line connecting � and . Determine the slope of the line between the interval Q�� R� For every two points .9 and .:, where .9 H .:, M.9� /9N and M.:� /:N form a straight line and create a _________ ___________. To determine the average rate of change for any function #M.N over an interval, we can use two points M.9� ��N and M.:� ��N�that lie on that interval. The process to find the slope of a linear function is:

( G /: E /9.: E .9

We can also use the slope formula to find the average rate of change over an interval Q�� R, where .9 G � and .: G .

BEAT THE TEST! 1.� The following graph is a piecewise function.

Which of the following statements are true about the graph? Select all that apply. �� The graph is increasing when the domain is

D I , I D�� The graph has exactly one relative minimum. �� The graph is increasing when , L D�. �� The graph is positive when , J �� The graph has no negative intervals. �� The range is - � K - I �� - L � . �� There is a relative minimum at M� N��

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Let’s Practice! 1.� Tom is jumping off the diving board at the city pool. His

height is modeled by the quadratic function �� % , H E� ,: D � , D �, where % , represents height above water (in feet), and , represents time after jumping (in seconds).

a.� Determine the average rate of change for the following intervals. ��

� !

!� "

b.� Compare Tom’s average rate of change over the

interval �� with his average rate of change over the interval R � !S. What does this represent in real life?

��N��O�

�N��O�

!�N��O�

Tom’s Jump into the Pool

��N��O��"��

Time (in seconds)

Hei

ght (

in fe

et)

Try It! 2.� Consider the table for the exponential function, ' , G A,

shown below.

Point 2 1 2

� � �

� �

� �

� �

a.� Determine the average rate of change over the

interval Q�� R. b.� Compare the average rate of change over the

interval �� with the average rate of change over the interval [�� R.


0

BEAT THE TEST!

1.� Suppose that the cost of producing ( radios is defined by � ( G �� C ��( D ��(8. Determine which of the following intervals has the greatest average rate of change for the cost to produce a radio.

�� Between � and � radios. �� Between � and � radios. �� Between � and �� radios. �� Between � and � radios.

3.� Determine the intervals that have the same average rate of change in the graph $ , G ,B below.


0

2.� Consider the absolute value function !M,N�and the step function " , in the graphs below.

��! , �"M,N

��

Which of the following is true about the rate of change of the graphs? �� The average rate of change for ! , over the interval

�� is greater than the average rate of change for "M,N over the interval Q$� %R.

�� The average rate of change for !M,N over the interval Q�� R is greater than the average rate of change for "M,N�over the interval�Q � (R.

�� The average rate of change for ! , �over the interval Q�� R is D7

8� �� The average rate of change for "M,N over the interval

Q � $R is D78�

�

�

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Section 3 – Topic 9 Transformations of Functions

The graph of #M.N is shown below.

�


0

Let’s Practice! 1.� For the following functions, state whether the independent

or dependent variable is being transformed and describe the transformation (assume % J �).

a.� ! , C %

b.� ! , D %

c.� !M, C %N

d.� !M, D %N

The following graphs are transformations of !M,N. Describe what happened in each graph. �! , C

! , D ��

! , C �

! , D �

Which graphs transformed the independent variable? Which graphs transformed the dependent variable?


0

2.� The following table represents the function " , �

2 /M2N D �� D� ��

The function # , G " , � Complete the table for # , �

2 /M52N 0M2N D� "MMD�NN D�� "MMD��NN � ��

Try It! 3.� The table below shows the values for the function ! , �

2 D D� � � .M2N � � �

Complete the table for the function D78 !M,N.

2 D45.M2N

D

D�

�

�


0

BEAT THE TEST! 1.� The graph of !M,N is shown below.�

Let " , G !M, D N and # , G ! , D . Graph "M,N and #M,N on the coordinate plane with ! , �

4.� The graph of�!M,N is shown below.�

Let " , G ! , C D . Graph "M,N on the coordinate plane with ! , �


2.� The table below shows the values for the function 'M,N.

2 D� D� � 1M2N � � � ��

Complete the table for the function 78 ' , D .

2 451 2 D 6

�

��

Test Yourself! Practice Tool

Great job! You have reached the end of this section. Now it’s time to try the “Test Yourself! Practice Tool,” where you can practice all the skills and concepts you learned in this section. Log in to Algebra Nation and try out the “Test Yourself! Practice Tool” so you can see how well you know these topics!

��

THIS PAGE WAS INTENTIONALLY LEFT BLANK


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BEAT THE TEST! 1.� The graph of !M,N is shown below.�

Let " , G !M, D N and # , G ! , D . Graph "M,N and #M,N on the coordinate plane with ! , �

4.� The graph of�!M,N is shown below.�

Let " , G ! , C D . Graph "M,N on the coordinate plane with ! , �


2.� The table below shows the values for the function 'M,N.

2 D� D� � 1M2N � � � ��

Complete the table for the function 78 ' , D .

2 451 2 D 6

�

��

Test Yourself! Practice Tool

Great job! You have reached the end of this section. Now it’s time to try the “Test Yourself! Practice Tool,” where you can practice all the skills and concepts you learned in this section. Log in to Algebra Nation and try out the “Test Yourself! Practice Tool” so you can see how well you know these topics!

��

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Section 4: Linear Equations, Functions, and Inequalities80

Section 4: Linear Equations, Functions, and Inequalities �

Topic 1: Arithmetic Sequences ............................................................................................................................................. 82

Topic 2: Rate of Change of Linear Functions ...................................................................................................................... 84

Topic 3: Interpreting Rate of Change and �-Intercept in a Real-World Context - Part 1 .............................................. 88

Topic 4: Interpreting Rate of Change and �-Intercept in a Real-World Context - Part 2 .............................................. 91

Topic 5: Direct and Inverse Variation ................................................................................................................................... 93

Topic 6: Introdution to Systems of Equations ......................................................................................................................... 97

Topic 7: Graphing Calculator Skills ....................................................................................................................................... 100

Topic 8: Finding Solution Sets to Systems of Equations Using Substitution and Graphing .............................................. 103

Topic 9: Using Equivalent Systems of Equations .................................................................................................................. 106

Topic 10: Finding Solution Sets to Systems of Equations Using Elimination ....................................................................... 110

Topic 11: Solution Sets to Inequalities with Two Variables .................................................................................................. 112

Topic 12: Finding Solution Sets to Systems of Linear Inequalities ....................................................................................... 116

Visit AlgebraNation.com or search "Algebra Nation" in your phone or tablet's app store to watch

the videos that go along with this workbook!


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The following Mississippi College- and Career-Readiness Standards for Mathematics will be covered in this section: A-CED.2 - Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-CED.3 - Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. �For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A-REI.5 - Given a system of two equations in two variables, show and explain why the sum of equivalent forms of the equations produces the same solution as the original system A-REI.6 - Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.10 - Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A-REI.11 - �Explain why the �-coordinates of the points where the graphs of the equations � � �� and � � �� intersect are the solutions of the equation �� ; find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, quadratic, absolute value, and exponential functions. A-REI.12 - �Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. F-BF.1 - Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-IF.3 - Recognize that sequences are functions whose domain is a subset of the integers. F-IF.6 - Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F-LE.2 - Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input/output pairs (including reading these from a table).

F-LE.5 - Interpret the parameters in a linear or exponential function in terms of a context. N-Q.1 - Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

N-Q.2 - Define appropriate quantities for the purpose of descriptive modeling.�S-ID.7 - Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.


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Section 4: Linear Equations, Functions, and Inequalities

Section 4 – Topic 1 Arithmetic Sequences

Let’s look at the following sequence of numbers: � ��

�� The “�” at the end means that this _______________ goes on forever.

�� and �are the actual ___________ of this

sequence. �� There are � terms in this sequence so far:

o� �is the �st term. o� � is the nd term. o� � is the _____ term. o� �� is the _____ term. o� is the _____ term.

This is an example of an arithmetic sequence.

�� This is a sequence where each term is the __________ of the previous term and a common difference,�!.

�

Let’s look at a way to find unknown terms:

Term Number

Sequence Term

Term Function Notation

� �8 "\�]

�9 � W U � "\] U �\�]

�: � W � U � W U � U � "\] �8 U �\]

� �; �� W � U � W U � U � U � "\�] �8 U �\]

� �< W �� U � W U � U � U � U � "\�] �8 U �\�]

�= � W U � W U � U � U � U � U � "\ ] �8 U �\�]

� �> W � U � W U � U � U � U � U � U � "\�] �8 U �\ ]

� �? � W U � W U � U � U � U � U � U � U � "\�]

� �@ � W � U � W U � U � U � U � U � U � U � U � "\�]

Write a general equation that we could use to find any term in the sequence. This is an explicit formula.

�� To solve for a term, you need to know the first term of the sequence and the difference by which the sequence is increasing or decreasing.

�


0

Let’s Practice! 1.� Consider the sequence �� V, V�, � ��

a.� Write an explicit formula for the sequence.

b.� Find the �nd term of the sequence. Try It! 2.� Consider the sequence �� .

a.� Write an explicit formula for the sequence.

b.� Find the next three terms of the sequence.

c.� Find the rd term of the sequence.

BEAT THE TEST!

1.� Yohanna is conditioning all summer to prepare for her high school’s varsity soccer team tryouts. She is incorporating walking planks into her daily training plan. Every day, she will complete four more walking planks than the day before.

Part A:� Yohanna starts with five walking planks on the first

day. Write an explicit formula that can be used to find the number of walking planks she completes on any given day.

Part B:� How many walking planks will Yohanna do on the �th day?

��

�


2.� The following figures were created with squares, where each side of the squares has a length of exactly one unit. �8 represents the area of the first figure, �9 represents the area of the second figure, and so on. Based on this pattern, what would be the rule for the area of the &PK figure when & X ��

�8 W � & W �

��9 W � ��& W

��: W � ��& W

�� M W � U �\& U �] �� M W � U �\& V �] �� M W � U \& U �] �� M W � U \& V �]

�

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Section 4 – Topic 2 Rate of Change of Linear Functions

Génesis reads � pages of The Fault in Our Stars every day. Zully reads � pages every day of the same book. Represent both situations on the graphs below using the same scales on the axes for both graphs.

Graph 1: Génesis’s Reading Speed

Graph 2: Zully’s Reading Speed


Aaron loves Cherry Coke. Each mini-can contains �� calories. Jacobe likes to munch on carrot snack packs. Each snack pack contains �� calories. Represent both situations on the graphs below using the same scales for both graphs.

Graph 3: Aaron’s Calorie Intake

Graph 4: Jacobe’s Calorie Intake

In each of the graphs, we were finding the rate of change in the given situation.

What is the rate of change for each of the graphs? Graph 1: ____________________ per _______________________ Graph 2: ____________________ per _______________________ Graph 3: ____________________ per _______________________ Graph 4: ____________________ per _______________________ This is also called the ________________________ of the line.

We can also find slope by looking at the KNJPML�OP�VKNJPML�OP�U� or

QORLQTP.

What is the slope of the following graph? What does the slope represent?


0

Let’s Practice! 1.� Consider the following graph.

a.� What is the rate of change of the graph?

b.� What does the rate of change represent?

2.� Freedom High School collected data on the GPA of various students and the number of hours they spend studying each week. A scatterplot of the data is shown below with the line of best fit.

a.� What is the slope of the line of best fit?

b.� What does the slope represent?


Try It! 3.� Sarah’s parents give her a �� allowance at the

beginning of each month. Sarah spends her allowance on comic books. The graph below represents the amount of money Sarah spent on comic books last month.

a.� What is the rate of change?

b.� What does the rate of change represent?

BEAT THE TEST!

1.� A cleaning service cleans many apartments each day. The following table shows the number of hours the cleaners spend cleaning and the number of apartments they clean during that time.

Apartment Cleaning

Time (Hours) � �

Apartments Cleaned � ��

Part A:� Represent the situation on the graph below.


Part B: The data suggest a linear relationship between the number of hours spent cleaning and the number of apartments cleaned. Assuming the relationship is linear, what does the rate of change represent in the context of this relationship?

�� The number of apartments cleaned after one

hour.�� The number of hours it took to clean one

apartment.�� The number of apartments cleaned each

hour.�� The number of apartments cleaned before

the company started cleaning.�

Part C: Which equation describes the relationship between the time elapsed and the number of apartments cleaned?

�� - Y ,�� - Y , W �� - Y ,�� - Y , W ��

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Section 4 – Topic 3 Interpreting Rate of Change and�1-Intercept

in a Real-World Context – Part 1 The T-Mobile ONE family plan includes four lines and unlimited data, calls, and texts for �� per month, including taxes and fees. There is an activation fee of ��. Define the variable and write a function that represents this situation. Represent the situation on a graph.

What is the slope of the line? What does the slope represent?


At what point does the line intersect the�--axis? What does this point represent? The --value of the point where the graph intersects the --axis is the --intercept. Let’s Practice! 1.� You saved � �� to spend over the summer. You decide

to budget � �� to spend each week.

a.� Define the variable and write a function that represents this situation.

b.� Represent the situation on a graph.

c.� What is the slope of the line? What does the slope represent?

d.� What is different about the slope of this line compared

to our earlier problem? Why is it different?

e.� What is the --intercept? What does this point represent?


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Try It! 2.� Consider the following graph.

a.� What is the slope of the line? What does the slope represent?

b.� What is the --intercept? What does the --intercept represent?

c.� Define the variables and write a function that represents this situation.

d.� What does each point represent?

Consider the three functions that you wrote regarding the cell phone plan, summer spending habits, and the community pool membership. What do you notice about the constant term and the coefficient of the ,�term?

�� The constant term is the ______________________. �� The coefficient of the , is the __________ or

_________________________. These functions are written in slope-intercept form. We can use slope-intercept form to graph any linear equation.

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��

The coefficient of , is the slope and the constant term is the --intercept ONLY if the equation is in slope-intercept form, - Y ', W "��


Section 4 – Topic 4 Interpreting Rate of Change and�1-Intercept

in a Real-World Context – Part 2 Let’s Practice! 1.� Graph - Y , W �.

2.� Consider the equation , W - Y �.

a.� How does this equation look different from slope-intercept form of an equation?

b.� Rewrite the equation in slope-intercept form.

c.� Identify the slope and --intercept.

d.� Graph the equation.


Try It! 3.� Graph the equation X�, X - Y X�.

BEAT THE TEST! 1.� Line ', ��, and �� are shown on the coordinate grid

below.

Which of the following statements are true? Select all that apply.

�� The slope of �� is equal to the slope of ��. �� The slope of �� is equal to the slope of line '. �� The slope of line ' is equal to

B@>B.

�� The slope of line ' is equal to ?CCA.

�� The )-intercept of line ' is�W�� X. �� Line '�represents a discrete function.


2.� The senior class at Elizabeth High School sold tickets to raise money for prom. The graph below represents the situation.

Part A: How much did one ticket cost?

Part B: How much money did the senior class have at the start of the fundraiser?

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Section 4 – Topic 5 Direct and Inverse Variation

The State Fair charges �� to ride the Ferris wheel. The graph below shows the relationship between cost (in dollars) and the number of rides.

Complete the table with values from the graph above.

0 1

�

�

Number of Rides

Co

st (

in d

olla

rs)

Ferris Wheel Cost per Ride


The graph and table show a relationship where the $- and %-values increase at the same rate. This is an example of direct variation because as the ________________ increases, the %-value increases or decreases at the same __________. �� The general form equation for direct variation is:

% J �$

�� In this equation, � is called the ___________ ____ __________________.

Suppose the Student Government Association at Jackson State University is hosting a graduation reception for their senior class. They decide to rent a ballroom for �� and split the cost evenly among all students who attend the event. The graph below shows the relationship between the number of students attending the event and the cost per student.

Fill out the table with information from the graph above:

0 1 �

�

��

��

Number of Students

Co

st p

er S

tud

en

t

Attendance and Cost per Student


The cost per student for the reception at Jackson State University is an example of an inverse variation; where the $-value ____________________ as the %-value ________________. �� The general form equation for inverse variation is:

% J � � �$

�� is the constant of proportionality for the variation. Let’s Practice! 1.� Describe the equation given below as a direct variation,

inverse variation, or neither. Justify your answer.

%$ J

2.� Create a graph below that models an inverse variation.

,

-


0

Try It! 3.� Correctly categorize the situations in the table below.

Situation Direct Variation

Inverse Variation

- Y X , o o Blue River Farms in Mount Olive, MS

charges �� per pound of blueberries

o o

- Y , o o -, Y o o

The cost per class at a gym that charges �� a month for unlimited

classes o o

,- Y � o o

BEAT THE TEST!

1.� Which of the following represent direct variation? Select all that apply. �� - Y X�,

�� VU Y

:;

�� , Y -

�� UV Y �

�� - Y @U

2.� Suppose the manager at the BMW manufacturing plant in

Greer, SC created a function to represent profit. He found that profit ()) varies directly with inventory (%). If the profit is �� when � cars from inventory are sold, which function below represents this relationship? �� ) Y O

>99

�� ) Y ��% �� ) Y >�999

O

�� ) Y ��%

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Section 4 – Topic 6 Introduction to Systems of Equations

A system of equations is a set of two or more equations. Consider the following system of equations. Line 1: , X - Y X Line 2:�, W - Y Graph the system of equations on the coordinate plane below.

Recall that a solution to a linear equation is any ordered pair that makes that equation a true statement.

What do you notice about the point ]X� ^? What do you notice about the point ]� �^? What do you notice about the point ]X� �^? What do you notice about the point ]� ^?


2.� Is there ever a time when a system of equations will not have a solution? If so, sketch an example.

Try It!

3.� Consider the following system of equations.

, X - Y � X, W - Y X�

a.� Sketch the graph of the system of equations.

b.� What can be said about the solution to this system of equations?

Let’s Practice! 1.� Consider the following system of equations made up of

Line 1 and Line 2. Line 1: , W - Y � Line 2: X�, X - Y X�

Complete the following sentences.

a.� The ordered pair ]X� ^ is a solution to

b.� The ordered pair ]� X^ is a solution to

c.� The ordered pair ]�� ^ is a solution to

o line 1� o line 2� o the system of equations.

o line 1. o line 2. o the system of equations.

o line 1. o line 2. o the system of equations.


0


�, W �- Y � , X - Y X

a.� Sketch the graph the system of equations.

b.� What is the solution to the system?

BEAT THE TEST!


, W - Y , X - Y X

Part A:� Sketch the graph of the system of equations.

Part B:� Determine the solution to the system of equations. Part C:� Create a third equation that could be added to

the system so that the solution does not change. Graph the line on the coordinate plane above.

� �


Section 4 – Topic 7 Graphing Calculator Skills

Imagine the Fulton, MS senior class goes on an annual trip. The transportation for the trip can be modeled by a system of equations, where , represents the maximum number of students in a van and - represents the maximum number of students in a bus. Below are the equations and graph that model this situation for last year and this year.

Last Year: �, W - Y � This Year: �, W �- Y ��

Transportation for the Trip

Approximate the number of students per van and per bus. How accurate is your approximation?

Van Passengers

Bus P

ass

enge

rs

,

-

2.� Two linear functions in a coordinate plane have no points of intersection. Which pair of functions does not intersect?

�� , W - Y and �, W �- Y �

�� , W - Y and �, W �- Y ��

�� , W - Y and - Y �� , X ��

�� , W �- Y � and - Y �� , X ��

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Let’s Practice! 1.� The system of equations that models the Fulton, MS trip

can be shown on a graphing calculator using different scales. Consider the graph below.


a.� Approximate the number of students per van and per bus.

b.� How accurate is your approximation? c.� Compare this approximation to your original

approximation.

Van Passengers

Bus P

ass

enge

rs

,

-

2.� The following graph models the same situation on a different scale.


a.� Approximate the number of students per van and per bus.

b.� How accurate is your approximation? c.� Does the change of scale on the axis help the

approximation become more accurate?

Van Passengers

Bus P

ass

enge

rs

,

-


BEAT THE TEST!

1.� The supply and demand equations for the Windbreaker 6000 Bicycle relate price per bicycle, ,, and the number of bicycles, -. The two equations are shown below.

Supply of bicycles: - Y � W ��, Demand for bicycles: - Y � X ,

The graph below shows the best range and scale on the axes for approximation.

Which of the following is the window range and scale represented in the graph? �� ,-axis: � X �, scale � and --axis: � X �, scale �� ,-axis: � X , scale and --axis: � X � �, scale � �� ,-axis: � X �, scale and --axis: � X ��, scale �� ,-axis: � X , scale � and --axis: � X ��, scale �

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Number of Bicycles (in thousands)

Pric

e p

er b

icyc

le (

in d

olla

rs)

,

-

Try It! 3.� Suppose the senior classes at Booneville High school and

Morton High School planned separate trips to New York City. The senior class at Booneville High rented and filled one van and six buses with �� students. Morton High rented and filled four vans and buses with �� students. The graph below models the equations for Booneville and Morton High School.


Determine a window range and scale for the ,- and -- axes that would provide the most accurate approximation.

Van Passengers

Bus P

ass

enge

rs

,

-


Section 4 – Topic 8 Finding Solution Sets to Systems of Equations

Using Substitution and Graphing We are able to use systems of equations to solve many real-world problems. One method of solving systems of equations is by graphing like we did in the previous video. Let’s Practice! 1.� Brianna’s lacrosse coach suggested that she practice

yoga to improve her flexibility. “Yoga-ta Try This!” Yoga Studio has two membership plans. Plan A costs �� per month plus �� per class. Plan B costs �� per month for unlimited classes.

a.� Define the variables and write two functions to

represent the monthly cost of each plan.

b.� Represent the two situations on the graph below.

c.� What is the rate of change for each plan?

d.� What does the rate of change represent in this situation?

e.� What do the --intercepts of the graphs represent?


Try It! 3.� Vespa Scooter Rental rents scooters for �� and��

per mile. Scottie’s Scooter Rental rents scooters for �� and �� per mile.

a.� Define the variables and write two functions to

represent the situation.

b.� Represent the two situations on the graph below.

c.� What is the rate of change of each line? What do they represent?

d.� What do the --intercepts of each line represent?

2.� Brianna is trying to determine which plan is more appropriate for the number of classes she wants to attend.

a.� When will the two plans cost exactly the same?

b.� When is Plan A the better deal?

c.� When is Plan B the better deal? We can also help Brianna determine the best plan for herself without graphing. Consider our two equations again. We want to know when the total costs would be equal.

�� Set the two plans equal to each other and solve for the number of visits.

�� This method is called solving by ____________________.


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It’s difficult to find the solution by looking at the graph. In such cases, it’s better to use substitution to solve the problem. 4.� Use the substitution method to help the renter determine

when the two scooter rentals will cost the same amount. a.� When will renting a scooter from Vespa Scooter Rental

cost the same as renting a scooter from Scottie’s Scooter Rental?

b.� Describe a situation when renting from Vespa Scooter Rental would be a better deal than renting from Scottie’s Scooter Rental.

BEAT THE TEST! 1.� Lyle and Shaun open savings accounts at the same time.

Lyle deposits �� initially and adds �� per week. Shaun deposits � �� initially and adds �� per week. Lyle wants to know when he will have the same amount in his savings account as Shaun.

Part A:� Write two equations to represent the amounts of

money Lyle and Shaun have in their accounts.

Part B:� Which method would you use to solve the problem, substitution or graphing? Explain your answer.

Part C:� After how many weeks of making the additional deposits will Lyle have the same amount of money as Shaun?

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Describe the result when we multiply either of the equations by some factor. Use this process to write an equivalent system.

Section 4 – Topic 9 Using Equivalent Systems of Equations

An ordered pair that satisfies all equations in a system is called the ________________ to that system. If two systems of equations have the same solution, they are called ___________________ systems. Let’s explore how to write equivalent systems of equations. Consider the following system of equations:

Equation A: , W - Y � Equation B: , X - Y �

The solution to this system is ] , X^. We can also see this when we graph the lines.


0

Consider the original system of equations again.

Equation A: ( Q ) S � Equation B: ( R ) S �

What is the resulting equation when we add the two equations in the system together? Graph the new equation on the same coordinate plane with our original system.

Algebraically, show that W , RX is also a solution to the sum of the two lines.

What is the resulting equation when we subtract the second equation from the first equation? Graph the new equation on the same coordinate plane with our original system.

Algebraically, show that ] , X^ is also a solution to the difference of the two lines.


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Graph Equation D on the same coordinate plane with our original system.

Algebraically, show that ] , X^ is a solution to the Equation D.

Let’s revisit the original system:

Equation A: , W - Y � Equation B: , X - Y �

Complete the following steps to show that replacing one equation by the sum of that equation and a multiple of the other equation produces a system with the same solutions. Create Equation C by multiplying Equation A by . Create Equation D by finding the sum of Equation B and Equation C.


0

Let’s Practice! 1.� Consider the following system, which has a solution of

]� ^. �� , and are non-zero real numbers.

�, W �- Y � �, W �- Y

Write two new equations that could be used to create an equivalent system of equations.

Try It! 2.� List three ways to write new equations that can be used to

create equivalent systems.

BEAT THE TEST!

1.� The system �, W �- Y ��, W �- Y � has the solution ],��^, where

�� and � are non-zero real numbers. Select all the systems of equations with the same solution.

�� X � , W � X � - Y � X � �, W �- Y �

�� ]� W �^, W ]� W �^- Y � W � ��, W �- Y �

�� , W �- Y � X��, X ��- Y X��

�� X � , W � X � - Y � X � �, W �- Y �

�� , W ]� W �^- Y � � W � , W �- Y � W �

��

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Section 4 – Topic 10 Finding Solution Sets to Systems of Equations

Using Elimination Consider the following system of equations:

, W - Y � , X - Y X

Write an equivalent system that will eliminate one of the variables when you add the equations. Determine the solution to the system of equations. Describe what the graph of the two systems would look like. This method of solving a system is called ____________________.

Let’s Practice! 1.� Ruxin and Andre were invited to a Super Bowl party. They

were asked to bring pizzas and sodas. Ruxin brought three pizzas and four�bottles of soda and spent �� . Andre brought five pizzas and two bottles of soda and spent �� a.� Write a system of equations to represent the situation.

b.� Write an equivalent system that will eliminate one of the variables when you add the equations.

c.� Solve the system to determine the cost of one pizza and one bottle of soda.


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Try It! 2.� Jazmin and Justine went shopping for back to school

clothes. Jazmin purchased three shirts and one pair of shorts and spent ��. Justine bought four shirts and three pairs of shorts and spent �� .

a.� Assuming all the shirts cost the same amount and all

the shorts cost the same amount, write a system of equations to represent each girl’s shopping spree.

b.� Use the elimination method to solve for the price of one pair of shorts.

BEAT THE TEST!

1.� Complete the following table.

Solve by Elimination: , X �- Y ��, W �- Y ��

Operations Equations Labels

, X �- Y ��, W �- Y �� Equation 1

Equation 2

X�, W �- Y X� New Equation 1

Multiply Equation by .

New Equation 2

X�, W �- Y X��, W �- Y ��- Y ��

Divide by �.

Solve for�,.

Write , and - as coordinates. ( , )

Solution to the system


2.� Which of the systems of equations below could not be used to solve the following system for , and -?

�, W �- Y � X, W �- Y X�

�� , W �- Y �

, X �- Y � �� , W �- Y �

X�, W �- Y X� �� , W - Y �

X�, W - Y X�� , W �- Y ��

X�, W �- Y X�

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Section 4 – Topic 11 Solution Sets to Inequalities with Two Variables

Consider the following linear inequality.

- \ , X Underline each ordered pair that is a solution to the above inequality.

]�� ^ ]X�X^ ]� ^ ]�� ^ ]�� ^ ]X�X�^ Plot each solution as a point in the coordinate plane.

Graph the line - Y , X in the same coordinate plane. What do you notice about the solutions to the inequality - \ , X and the graph of the line - Y , X ?


Let’s Practice! 1.� The senior class is raising money for Grad Bash. The

students’ parents are donating cakes. The students plan to sell entire cakes for �� each and slices of cake for �� each. If they need at least � ��, how many of each should they sell?

a.� List two possibilities for the number of whole cakes and

cake slices students could sell to reach their goal of raising at least � ��.

b.� Write an inequality to represent the situation.

c.� Graph the region of the solutions to the inequality.

d.� What is the difference between the ordered pairs that fall on the line and the ones that fall in the shaded area?

e.� What does the ,-intercept represent?


Try It! 2.� The freshman class wants to include at least � people in

performances at a pep rally. Each skit will have people, and each dance routine will feature people.

a.� List two possible combinations of skits and dance

routines.

b.� Write an inequality to represent the situation.

c.� Graph the region of the solutions to the inequality.

d.� What does the --intercept represent?


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BEAT THE TEST!

1.� Coach De Leon purchases sports equipment. Basketballs cost �� each and soccer balls cost �� each. He has a budget of � ��. The graph shown below represents the number of basketballs and soccer balls he can buy given his budget constraint.

Part A: Write an inequality to represent the situation.

Part B: Determine whether these combinations of basketballs, ", and soccer balls, *, can be purchased within the budget.

. Y 6 / Y 4

. Y 3 / Y 5

. Y 2 / Y 8

. Y 8 / Y 2

. Y 5 / Y 7

Yes � � � � � No � � � � �

��

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Section 4 – Topic 12 Finding Solution Sets to Systems of Linear Inequalities

Juan must purchase car insurance. He needs to earn at least � �� a week to cover the payments. The most he can work each week is � hours because of football practice. Juan can earn �� per hour mowing yards and �� per hour washing cars. The system �, W - \ �

, W - [ �� represents Juan’s situation.

Define the variables. The graph below depicts Juan’s situation. Interpret the graph and identify two different solutions.

Let’s Practice! 1.� Bristol is having a party and has invited � friends. She

plans to purchase sodas that cost � �� for a -pack and chips that cost �� per bag. She wants each friend to have at least two sodas. Bristol’s budget is ��

a.� Write a system of inequalities to represent the

situation.

b.� Graph the region using boundary lines to show the solutions to the inequality.

c.� Name two different solutions for Bristol’s situation.

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Continuity of Learning Assignments Essential …...Continuity of Learning Assignments Grade: _9-12___ Subject: _Math_____ Days 1-2 Days 3-4 Days 5-6 Days 7-8 Days 9-10 Essential Activity