Download Closure Property

Week 10
Name______________________________________________________ Period: 2&6 3&7 4&8
Section 3 Topic 5 (A-APR.1.1) – Closure Property (Polynomials)
Standard(s): MAFS.912.A-APR.1.2 - Understand that polynomials form a system analogous to the integers, namely, they are closed
under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Essential Question: What is the difference in the statement, “the real numbers are closed under non-zero division” and this
statement, “the real numbers are closed under division”?
Key Vocabulary: polynomials, integers, rational numbers, irrational numbers, closed
Bellwork
Vocabulary
Integers- ____________________________________________________________________________________________________
Rational Numbers- ____________________________________________________________________________________________
Real Numbers - _______________________________________________________________________________________________
Polynomial- __________________________________________________________________________________________________
Guided Notes
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.
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Are integers closed under addition? Justify your answer.
Are irrational numbers closed under multiplication? Justify your answer.
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.
Polynomials are closed under division.
I Do
We Do
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BEAT THE TEST!
Choose from the following words and expressions to complete the statement below.
The product of 5𝑥 4 − 3𝑥 2 + 2 and _______________________ illustrates the closure property because the
_______________ of the product are ____________________ , and the product is a polynomial.
Introduction to Functions
Closure Property
Classwork/Independent Practice
1. Mr. Steffens claims that the closure properties for whole numbers are closed when dividing whole numbers.
Mr. Slater claims that the closure properties for whole numbers are not closed when dividing whole
numbers. Who is correct? Explain your answer.
2. 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
Rational numbers are closed under addition.
True
False
Example:
Answer Choice
Statement B
Rational numbers are closed under subtraction.
True
False
Example:
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Name______________________________________________________ Period: 2&6 3&7 4&8
Answer Choice
Statement C
Rational numbers are closed under multiplication.
False
True
Example:
Answer Choice
Statement D
Rational numbers are closed under division.
False
True
Example:
3. 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...}








10
8
6
4
2
{... − 2 , −2, −2, −2, − 2 ...}
Home Learning
4. Consider the following polynomials.
𝑎𝑏 2 + 3𝑎𝑏 + 8𝑎2
−5𝑎𝑏 2
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.
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Week 10
Name______________________________________________________ Period: 2&6 3&7 4&8
Section 3 Topic 6 (F-BF.1.1)– Real-World Combination and Composition of Functions
Standard(s): MAFS.912.F-BF.1.1b.c. - Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations. 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.
c. 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.
Essential Question: How can the operations of real numbers be applied to functions to create real-world combinations and
compositions?
Key Vocabulary: function notation, combination, composition
Bellwork
1. Given 𝑝(𝑥) = 2𝑥 2 + 4 and 𝑞(𝑥) = (−3𝑥 2 + 4)
Find 𝑝(𝑥) • 𝑞(𝑥)
2. Let ℎ(𝑥) = 2𝑥 2 + 𝑥 − 5 and 𝑔(𝑥) = −3𝑥 2 + 4𝑥 + 1
Find ℎ(𝑥) + 𝑔(𝑥)
Guided Notes:
Describe the difference you see in the combinations and compositions of functions:
Combinations
Compositions
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Do
For 1-3: Let 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = 3𝑥, 𝑎𝑛𝑑 ℎ(𝑥) = 𝑥 2 + 1
1. 𝑔(ℎ(𝑥)) =
We Do
For 1-3: Let 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = 3𝑥, 𝑎𝑛𝑑 ℎ(𝑥) = 𝑥 2 + 1
1. 𝑓(𝑔(𝑥)) =
Input:______
Input:______
Output:____
Output:____
For 1-3: Let 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = 3𝑥, 𝑎𝑛𝑑 ℎ(𝑥) = 𝑥 2 + 1
2. ℎ(𝑓(𝑥)) =
For 1-3: Let 𝑓(𝑥) = 2𝑥 − 1, 𝑔(𝑥) = 3𝑥, 𝑎𝑛𝑑 ℎ(𝑥) = 𝑥 2 + 1
2. 𝑓(ℎ(𝑥)) =
Input:______
Input:______
Output:____
Output:____
3. ℎ(𝑔(2)) =
3. 𝑔(𝑓(−3)) =
Input:______
Input:______
Output:____
Output:____
4. Charlotte is selling Hot Fires to raise money for her college
textbooks. The cost of each bag of Hot Fries is $2. There is a
$20 fee to sell them at the school. She plans to sell the Hot
Fries for $4.
Part A – Define the variable.
4. The freshman class is selling t-shirts to raise money for a field
trip. The cost of each custom designed t-shirt is $8. There is a
$45 fee to create the design. The class plan to sell the shirts for
$12.
Part A – Define the variable
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Part B – Write a cost function.
Part B – Write a cost function.
Part C – Write a revenue function.
Part C – Write a revenue function.
Part D – Write a profit function.
Part D – Write a profit function.
5. You work forty hours a week at a furniture store. You receive
a $220 weekly salary, plus a 3% commission on sales over
$5000. Assume that you sell enough this week to get the
commission. Given the functions 𝑓(𝑥) = 0.03𝑥 and
𝑔(𝑥) = 𝑥 – 5000.
5. You work forty hours a week at a car dealership. You receive
a $425 weekly salary, plus a 7% commission on sales over
$12000. Assume that you sell enough this week to get the
commission. Given the functions ℎ(𝑥) = 0.07𝑥 and
𝑔(𝑥) = 𝑥 – 12000.
Part A: What does 𝑔(𝑥) represent?
Part A: What does 𝑔(𝑥) represent?
Part B: What does 𝑓(𝑥) represent?
Part B: What does 𝑓(𝑥) represent?
Part C: Write a composition function (𝑓 ° 𝑔)(𝑥) to represent
the amount of money you earn on your commission?
Part C: Write a composition function (𝑓 ° 𝑔)(𝑥) to represent
the amount of money you earn on your commission?
Part D: How much profit would you make on sales of $12,000?
Part D: How much profit would you make on sales of $25,000?
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Week 10
Name______________________________________________________ Period: 2&6 3&7 4&8
Introduction to Functions
Real-World Combinations and Compositions of Functions
Classwork/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 $𝟏. 𝟓𝟎 and the florist charges a delivery fee of $𝟐𝟓. The class
plans to sell the roses for $𝟓. 𝟎𝟎 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 𝟐𝟓𝟎𝟎 t-shirts for $𝟏𝟎. They
consider raising the t-shirt price in order to earn more profit. For each $𝟏 increase, they will sell 𝟏𝟎𝟎 fewer tshirts. Let 𝒙 represent the number of $𝟏 increases.
a. Write a function, 𝑪(𝒙), to represent the cost of one t-shirt based on the number of increases.
b. Write a function, 𝑻(𝒙), to represent the number of t-shirts sold based on the number of increases.
c. Write a revenue function, 𝑹(𝒙), for the t-shirt sell that could be used to maximize revenue.
Home Learning
1. Anna gets paid $𝟖. 𝟕𝟓/hour working as a barista at Coffee Break. Her boss pays her $𝟗. 𝟎𝟎/hour for creating
the weekly advertisement signs. She works a total of 𝟐𝟓 hours each week.
a. Let 𝒙 represent the hours that Anna works each week as a barista. Write a function 𝒉(𝒙) to represent the
amount of money that Anna earns working as a barista.
b. Write a function, 𝒇(𝒙) to represent the hours Anna works creating the signs.
c. Let 𝒔 represent the number of hours that Anna works creating the signs. Create a function 𝒈(𝒔) to
represent the money Anna earns creating the signs.
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d. Find 𝒈(𝒇(𝒙)). 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 𝒙 represents the number of
hours she works as a barista.
Section 3 Topic 7 – Key Features of Graphs of Functions – Part 1
Standard(s): MAFS.912.F-IF.2.4 - For 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative;
relative maximums and minimums; symmetries; end behavior; and periodicity.
Essential Question: What are key features of graphs?
Key Vocabulary: vertical line test, x-intercept, y-intercept, linear, non-linear, domain, range, evaluate
Bellwork
1.
2. In your own words, describe what a continuous function
means.
3. In your own words, describe what a discrete function means.
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Guided Notes
Label everything you can about the graph below. This will require you to use prior knowledge. Always strive to use appropriate and
correct math vocabulary.
Find each value below:
𝑓(0) =
𝑓(−2) =
𝑓(3) =
Domain: ____________________________________________________________________________________________________
Range: _____________________________________________________________________________________________________
X- intercept: _________________________________________________________________________________________________
Y-intercept: __________________________________________________________________________________________________
Linear: ______________________________________________________________________________________________________
I Do
Use the vertical line test to determine whether the graph is a
function.
You Do
Use the vertical line test to determine whether the graph is a
function.
Explanation:
Explanation:
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Week 10
Name______________________________________________________ Period: 2&6 3&7 4&8
Determine if the graph is a function. Then find the domain and
range of the graph.
Find the domain and range of the graph.
Function?
Domain:
Function?
Range:
Domain:
Range:
Find the domain and range of the graph. Then find the
intercepts.
Domain:
Find the domain and range of the graph. Then find the
intercepts.
Domain:
Range:
Range:
x-intercept(s):
x-intercept(s):
y-intercept(s):
y-intercept(s):
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Domain:
Range:
Function?
x-intercepts:
y-intercepts:
Discrete or Continuous?
𝑓(−1) =
) = −1
𝑓(
Introduction to Functions
Key Features of Graphs of Functions – Part 1
Classwork/Independent Practice
1. Determine if the following graphs are functions. Determine the domain, range, y-intercepts and yintercepts.
a.
Function?
b.
Function?
c.
Domain:
Domain:
Range:
Range:
x-intercept(s):
x-intercept(s):
y-intercept(s):
y-intercept(s):
Function?
d.
Function?
Domain:
Domain:
Range:
Range:
x-intercept(s):
x-intercept(s):
y-intercept(s):
y-intercept(s):
2. 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.
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3. 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.
4. Sketch the graph of a relation that is a function.
5. Sketch the graph of a relation that is NOT a function.
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Name______________________________________________________ Period: 2&6 3&7 4&8
6. Use the word bank to complete the sentences below.
𝑥 −coordinate
𝑦 −coordinate
𝑥 −intercept
𝑦 −intercept
solution
a. The ____________________ of a graph is the location where the graph crosses the 𝑥 −axis.
b. The ____________________ of a graph is the location where the graph crosses the 𝑦 −axis.
c. The ____________________ of the 𝑦 −intercept is always zero.
d. The ____________________ of the 𝑥 −intercept is always zero.
e. The 𝑥 −intercept is the ______________ to a function or group.
Home Learning
The path of a golf ball
a. The graph is (circle one) linear/nonlinear.
b. Is the graph a function? Explain.
Elevation (ft)
7. The graph to the right represents the path of a golf ball.
c. What is the 𝑦 −intercept and what does the
𝑦 −intercept represent?
Distance (ft)
d. What a solution to this graph and what does it represent in this situation?
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