Guess Who…

• Rules– Yes or No Questions ONLY– One question per person

• Whiteboards: Brainstorm questions you will ask to determine who you are

• Predict: How many questions until you figure out who you are?

Play…

• Once you figure out who you are sit back down

• Erase whiteboards and answer:– Did it take more or less questions than you

thought? Explain WHY this happened

Guess Who…What is the equation of this polynomial?

• Create a PLAN: How would you go about determining the equation?

Unit 3- Polynomials and Quadratics in the Real Number System

3A-I can identify key features of a polynomial (graphically &

algebraically) and explain the connection between roots, factors,

multiplicity, degree, and the stretch/compress factor in writing

equations of and graphing polynomials.

I. What’s in a Polynomial Name?A. Vocabulary

① A sum of monomials or a single monomial is considered a polynomial

② Polynomials can be written in two forms:i. Standard Form-Written with terms in descending order by

degree

ii. Factored Form-Written with factors instead of terms

③ Polynomials are classified by their degree (exponent with the highest value), their terms, or when possible both

B. Classifying Polynomials1) By Degree

0=Constant1=Linear2=Quadratic3=Cubic4=Quartic

2) By Terms1 term=Monomial2 terms=Binomial3 terms=Trinomial4 or more terms=Polynomial

C. Sketching Polynomials

– Use the chart and graphing calculators to sketch each polynomial in the correct “spot”

1) What is the connection between the degree of the polynomial and the roots of the polynomial?

2) Explain the difference between the graphs of polynomials with even degrees versus odd degrees.

3) Sketch what the graph of f(x)= -8x7 would look like. DEFEND in words that your sketch is “correct”.

D. Examples①Describe

everything you possibly can about the polynomial you see

D. Examples②Describe

D. Examples③Describe

GUESS WHO…• Player A describes to Play B • Player B tried to draw the polynomial• Player A describes “missing pieces” needed

until Player B’s graph is good

• Switch roles

HOMEWORK

• Page 318 (1-10)

• Write out the end behavior the way we have done in class!!!

• Or work on concept map from Unit 2

• Create a PLAN: What ideas do you have now?

• Create a PLAN: What ideas do you have now?• Together: What do you think we NEED to find the

equation?

Model Card Sample

Model Card Must Address These Issues…Quadratic Problem Table Problem Pizza Problem

1) Explain in words what f(b)=f(3) means

2) Explain steps taken to solve for f(b)

3) What does the minimum have to do with the problem?

4) Use correct vocabulary in explanation

1) Clearly explain what the table of values represents

2) What is the difference between function f and function g?

3) Explain steps taken to solve for g(2)

4) Use correct vocabulary in explanation

1) Clearly explain what the C(t) formula means in words

2) Explain what “best price” means

3) Explain how to calculate C(9) and C(8)

4) Interpret the meaning of C(9)-C(8)

Exit Post-It On HomeworkPICK ONE and sketch the polynomial

f(x)=-5x3+7x2-10x

f(x)=-8x4+5x3-2x2+7x-1

Write the end behavior of the polynomial

II. Polynomial Equations Graphs

A. Definitions① Roots=X-Intercepts=Zeros=Solutions② Polynomials can be written in two forms:

standard form and factored form

③ Factors are the “pieces” used to build/break down a polynomial function

B. Relationship between Roots and Factors of a Polynomial

B. Relationship between Roots and Factors of a Polynomial EXPLORE with your graphing calculator ANSWER (on your paper): What pattern do you

notice between the function in factored form and the roots of the function?

Keep Going After you have answered the question on the paper…

B. Relationship between Roots and Factors of a Polynomial EXPLORE with your graphing calculator ANSWER: What pattern do you notice between the

function in factored form and the roots of the function?

TOGETHER (whiteboards): Based on your pattern… If a root of a polynomial was M what would a factor of

the polynomial be? If a factor of a polynomial was (x+c) what would a root

of the polynomial be? If a factor of a polynomial was (Ax+c) what would a

root of the polynomial be? EXPLAIN YOUR REASONING!!!

B. Relationship between Roots and Factors of a Polynomial If a polynomial has a root (x-intercept) of M, then

the polynomial has a factor of (X-M)

C. Key Features of Polynomials– ???– ???– ???– ???– ???– ???– ???– Stretch/Compress Factor (A)

– Multiplicity

D. Relationship between ODD and EVEN Multiplicity Factors

EXPLORE with your graphing calculator ANSWER (on your paper): What pattern do you

notice between the function in factored form and the GRAPH of the function?

Keep Going After you have answered the question on the paper…

Homework

• Level A Pg. 323 (13, 14, 16-18)• Level B Pg. 323 (19-20, 47-50)• Level C Pg. 353 (15, 37, 39, 58)

• EVERYONE PICK TWO OF Pg. 323 (57, 59, 61, or 64)

EXPLORE with your graphing calculator ANSWER (on your paper): What pattern do you notice between the function in factored form

and the GRAPH of the function?

TOGETHER (without a calculator): Sketch a graph of the following

g(c)=8(c+2)4(c-9)3

h(t)=-5(t-10)6(t-8)2

A polynomial function with roots at x=-4 with a multiplicity of 3, x=0 with a multiplicity of 5, and x=7 with a multiplicity of 6

EXPLAIN YOUR REASONING!!!

i. When factors/roots appear more than once in a polynomial function this is called multiplicity

ii. EVEN Multiplicities will look like…

iii. ODD Multiplicities will look like…

iv. Sketch the graph of g(c)=8(c+2)4(c-9)3

V. Sketch the graph of h(t)=-5(t-10)6(t-8)2

vi. Sketch the graph of a polynomial function with roots at x=-4 with a multiplicity of 3, x=0 with a multiplicity of 5, and x=7 with a multiplicity of 6

Homework

• Level A Pg. 323 (21-28) Write in FACTORED FORM

• Level B Pg. 323 (29, 30, 33-36, 51-53)• Level C Pg. 353 (15, 37, 39, 58)

• EVERYONE PICK THE OTHER TWO OF Pg. 323 (57, 59, 61, or 64)

III. The Stretch/Compress Factor

A. Sketch a graph of the following three functions and explain IN WORDS the difference you see in the graphsf(x)= (x-3)2(x+1)3 f(x)=8(x-3)2(x+1)3 f(x)= ¼ (x-3)2(x+1)3

B. The Stretch/compress factor is the “a” in factored and standard form of a polynomial.

When A is…

The polynomial is

C. Examples-Write the equation in factored form for the function below

(0, -120)

(1, -10.5)

(4.781, -4.034)

Sideways

• Level A Equation in Factored form when stretch compress factor is a=1

• Level B Equation in Factored form when stretch compress factor is a≠1

• FIND Answer then move SIDEWAYS

• Create a PLAN: What ideas do you have now?• Write down everything you can about this

polynomial (analyze the polynomial)

• Create a PLAN: What ideas do you have now?• In teams on big whiteboards set up Model Card

(make sense of problem and step to solving)

Guess Who…

• Sort the polynomials into categories– No polynomial can be on it’s own

• Name your categories

Guess Who…• Sort the polynomials into categories– No polynomial can be on it’s own

• Write each polynomial onto a post-it

• Together sort the polynomials into at least four categoriesNAME each category

• No polynomial can “fit” into more than one category

Guess Who…• Sort the polynomials into categories

– No polynomial can be on it’s own

• Write each polynomial onto a post-it

• Together sort the polynomials into at least four categoriesNAME each category

• Record your new thinking

Learning Target 3B

I can create equivalent expressions/functions of polynomials by applying various factoring techniques and explain the connections between the algebraic and graphical form of the polynomial.

NOTES FROM SUB DAY GO HERE

I. Converting Factored Form into Standard Form

One last exampleConvert f(x)=-3(x-3)2(x+1) to standard form

II. Factoring Polynomials

PRIOR: In Algebra I you learn how to factor all of the following:

①x2-10x+21②6x3-12x2+36x③49b2-c2

④3x2+4x-4

PRIOR: In Algebra I you learn how to factor all of the following

①x2-10x+21②6x3-12x2+36x③49b2-c2

④3x2+4x-4⑤16y2-1⑥4m2x+7m2y-8x-14y⑦6x4y2+8x2y-8

1) Can you factor each polynomial?

2) Can you name the factoring method you are using?

3) Try the level C Algebra I factoring questions 5-7

Homework

• Watch a video OR read about factoring:– Factoring greatest common factor from polynomials– Factoring difference of squares– Factoring quadratic trinomials when a=1– Factoring quadratic trinomials when a≠1

• Pick TWO that you need to review Bring in EVIDENCE of having watched the video or print and annotate the page from on-line

A. Special Factoring Patterns① Sum of Cubes a3+b3=(a+b)(a2-ab+b2)

② Difference of Cubes a3-b3=(a-b)(a2+ab+b2)

③ Difference of Squares a2-b2=(a-b)(a+b)

THINK: 13= 43= 73= 103= 23= 53= 83= 113=33= 63= 93= 123=

B. Examples① Factor x3+27

②Factor -27u3-125

B. Examples③ Factor 250x4+128x

④Factor a3b3-343c3

B. Examples⑤ Factor 64x6-8y12

⑥Factor b6+216c3u15

ASK:Is there a GCF?

(greatest common factor?

Quadratic/Quartic or Cubic?

Factor out the Greatest Common

Factor FIRST

Binomial or Trinomial?

Sum or Difference of Cubes?

Sum or Difference of Squares?

ax2+bx+c

Can’t Factor any Further

Easy MENTAL “X” Method

“X” Method (or box

method)

Can’t Factor

Factor by Grouping?

C. Factoring Concept Map

Factoring Polynomials

• Level AFactor

• Level BFactor

• Level CFactor

Factoring Polynomial

• Level A

• Level B

• Level C

Guess Who…

• How has your thinking changed between the first sorting activity to now?

Create a NEW “Unit 2” Concept Map

• NEW PIECE OF PAPER!!!• Model CardsAdd to Interpreting Functions

(Unit 2) Map?• Add onto the map everything you can from

Polynomial Functions (Unit 3 )

• Is functions still your “center”?• FIX bad connections

#STEALING #CONSIDER• #STEALING– I like your connection between _____ and _____

because…– Your visual (graph, chart, etc.) really helped me

understand …– I never thought about ______ in that way; now I

understand …• #CONSIDER– A visual (graph, chart, etc.) would really help you explain

_________ because…– I didn’t really understand your connection between

_______ and ______ because…– I think you need to fix _______ because…

Guess Who Is/Isn’t…

Leotis claims that 5, 4, -3, and -5 are all roots of the function f(x)=x4-7x3-13x2+175x-300. Create as many mathematical arguments as possible to explain whether or not Leotis is correct.

Learning Target 3CI can explain the relationship between roots, factors, and solving polynomials and use this relationship to determine when to use factoring, polynomial long

division and/or the Remainder Theorem to solve polynomials.

I. “Solving” Polynomials A. When solving a polynomial you are being asked

to find the solutions to the polynomial• Solutions=_________=__________=_________• The __________ of a polynomial is what tells me how

many solutions the polynomial has• Roots and factors of polynomials are _________ of

each other

I. “Solving” Polynomials A. When solving a polynomial you are being asked

to find the solutions to the polynomial• Solutions=_________=__________=_________• The __________ of a polynomial is what tells me how

many solutions the polynomial has• Roots and factors of polynomials are _________ of

each other

THINK: When trying to find the roots of functions why do we always make f(x)=0? (Answer with a mathematical argument)

WHITEBOARDS

I. “Solving” Polynomials A. When solving a polynomial you are being asked to find the

solutions to the polynomial• Solutions=_________=__________=_________• The __________ of a polynomial is what tells me how many

solutions the polynomial has• Roots and factors of polynomials are _________ of each other

THINK: When trying to find the roots of functions why do we always make f(x)=0? (Answer with a mathematical argument)

1) Graph m(b) and label as many key features of the graph as possible

B. Examples

2) Graph g(c) and label as many key features of the graph as possible

3) Graph f(x) and label as many key features of the graph as possible

Partner A: Graph f(x) and label as many key features of the graph as possible

Partner B: Graph f(x) and label as many key features of the graph as possible

LEVEL A: Graph f(x) and label as many key features of the graph as possible

LEVEL B: Graph f(x) and label as many key features of the graph as possible

LEVEL C: Graph f(x) and label as many key features of the graph as possible

Work on the Whiteboards…If your graph does not look like one below you made

a mistake!

Homework• Level A/B Pg. 336 (2, 3, 5, 8, 9, 12-14, 51, 63)– Solve AND GRAPH like we did in notes today

• Level C Pg. 336 (21-28, 48, 54)– Solve AND GRAPH like we did in notes today

• EVERYONE Pg. 338 (68, 69)

• Opportunity to improve: GRADED Exit Ticket Next Week

II. Dividing PolynomialsA. One way to determine if a polynomial is a factor

of another polynomial is to divide themB. When dividing polynomials (just like numbers)• If the remainder = 0 then the polynomial is a factor• If the remainder ≠ 0 the the polynomial is not a factor

PRIOR: Is 51 a factor of 14,076? Explain how you know.

WHITEBOARDS

PRIOR: Is 51 a factor of 14,076? Explain how you know.

C. Examples① Is (5x+1) is a factor of 10x3+37x2+37x+6? Justify

your answer.

C. Examples② What is the remainder when 6x3+7x2-1 is divided

by 2x+1? Explain what that remainder means.

C. Examples③ Simplify x3-x2-14x+24

Polynomial Long Division

• Level ASimplify

• Level BSimplify

• Level CSimplify

Polynomial Long Division

• Level A

• Level B

• Level C

Quick Check FYI

①Let and

then what does equal?

②What is the remainder when is divided by ? Explain what the remainder means

P(x)Q(x)

①3x2+4x+10+

②The remainder equals zero; thus x-1 is a factor of -3x3+4x-1 and therefore x=1 is a root of -3x3+4x-1

III. The Remainder & Factor TheoremsA. Remainder Theorem• If a polynomial f(x) is divided by x-k, then the

remainder is r = f(k)

B. Factor Theorem• A polynomial f(x) has a factor x-k if and only if

f(k) = 0

III. The Remainder & Factor TheoremsC. Synthetic Division can only be used when

testing/dividing a linear polynomial with a=1

D. Synthetic division is a way to quickly test if a x-value is/is not a root of the given polynomial

E. Examples① Divide f(x)=x3+2x2-6x-9 by (a) x-2 and (b) x+3.

Using the division answer explain what f(2) and f(-3) are equal to.

E. Examples② Given that f(-3)=0 factor

p(x)=2x3+11x2+18x+9

Quick Check FYI

①Sketch the graph of p(x)=2x3+11x2+18x+9. Include as many key features as possible.

E. Examples③ One zero of m(n)=n3-2n2-9n+18 is n=2.

Find the other zeros of the function.

Homework Options• Long Division– Pg. 356 (4-7), (21-26)

• Synthetic Division– Pg. 356 (8-11), (27-38)

• Learning Target 3C Finding all zeros of a function– Pg. 357 (39-54)

• Understanding– Pg. 357 (55, 56, 61, 65)

Guess Who…

Won the egg launch contest.

Create a mathematical argument to defend which team you think won the contest.

Final is Coming…• 22 days and counting

• THREE PARTS1. Fluency (All Multiple Choice, must do all)2. Understanding (Three Create mathematical argument, pick

one)3. Model Card & Mind Map created in class*

• Goal: 4 to 5 days of stations review in class– At home…use class website to prepare– Start making a mind map using your notes– Annotate your own notes; use post-its to mark things you

have questions on

Learning Target 3D: I can analyze a quadratic problem, within context, formulate a mathematical model (equation, table, graph etc.), compute an

answer or rewrite the expression to reveal new information, and interpret the meaning

of my results.

I. Special Formulas For QuadraticsA. In general polynomials have two forms

1. __________________________2. __________________________

B. Quadratics, a special type of polynomial, have other formulas associated with some of their key features that are also important sources of information (Quadratics also have a third form!)

C. The important part is determining what is needed when solving particular Quadratic problems; for example…

What is the maximum height Jason reaches? How many seconds does Jason spend in the air? Create a mathematical argument to defend your answers.

②A projectile is launched from the ground with an initial velocity of 35 ft/s. Using the function h(t)=-16t2+vt+h0, determine when the projectile will first reach a height of 8 feet and how many seconds later it will again be at 8 feet. If there is a tree 10 feet tall in the middle of the projectile’s path will the projectile be able to clear the tree?

I’m Sorry

Try These…Use Calculator/WhiteboardsY=-16t2+30t+35 x=-4 and x=-5F(x)=-16x2+30x+60 t≈-4.74 and t≈1.4Y=1.2t2+4t-8 X=4F(x)=x2-8x+16 t≈-0.8 and t≈2.7x2-10=-3x t≈-1.2 and t≈3.1

1)Twice the square of a positive number increased by 3 times the number is 14. Find the number.

2)The length of a rectangular garden is 2 feet less than 3 times its width. Find the length and width if the area of the garden is 21 square feet.

②A projectile is launched from the ground with an initial velocity of 35 ft/s. Using the function h(t)=-16t2+vt+h0, determine when the projectile will first reach a height of 8 feet and how many seconds later it will again be at 8 feet. If there is a tree 10 feet tall in the middle of the projectile’s path will the projectile be able to clear the tree?

I’m Sorry

3) The function N(t)=0.0054t2-1.46t+95.11 can be used to estimate the average number of years of life expectancy remaining for a person of age t years where 30 ≤ t ≤ 100. What is your remaining life expectancy? If a person has a remaining life expectancy of 14.3 years, estimate the age of the person.

Framing

– Annotate, mind map, create a plan, execute your plan, …

– 10 minutes on your own

Framing

– 10 minutes on your own– Write a sentence what are you doing and why are

you doing that?

Framing

you doing that?– 10 minutes together

Framing

you doing that?– 10 minutes together – Write a sentence what are you doing and why are

you doing that?

Homework

253 (FLUENCY 28-30, 54)

UNDERSTANDING #55 Must write a mathematical argument to defend your answer choice

II. Vertex Form of QuadraticsA. Formulas

① The Vertex Form of a Quadratic is f(x)=a(x-h)2+k • “h” represents the horizontal shift• “k” represents the vertical shift• Vertex (h,k)• Axis of Symmetry x=h

B. Examples① Sketch the graph of f(x)= -½(x-4)2+10.

Include the y-intercept and at least two other points in your sketch.

B. Examples② Sketch the graph of f(x)= 5(x+7)2+4.

Include the y-intercept and at least two other points in your sketch.

B. Examples③ The graph g(x)= ¼(x-4)2+9 has been shifted

to g’(x)=-¼(x+2)2+11. If point P (0,13) and point Q (-4, 25) where on g(x) where would points P’ and Q’ be located?

Homework

Level A Pg. 260 (21-26, 36-41, 43-46)

Level B Pg. 260 (13-20, 40-41, 47-50, 74)

Level C Read Example 4 on page 259 then try Pg. 260 (27-33, 51-53)

EVERYONE Pg. 260 (42, 69, 77)

B. Examples③ The graph g(x)= ¼(x-4)2+9 has been shifted

to g’(x)=-¼(x+2)2+11. If point P (0,13) and point Q (-4, 25) where on g(x) where would points P’ and Q’ be located?

1) As a group work together to create a model card of how you are making sense of the problem and then the steps you are taking to solve the problem.

2) When finished begin reading how to do Quadratic Regression on your calculator by following along with the steps/example given to you

3) When finished create a quadratic model that best represents the data to the right

To the left is a table of values generated by a quadratic function.

What key features can you determine from this

table?EXPLAIN how you know.

X f(x)=y-1 -150 01 92 123 94 0

III. Quadratic RegressionA. Definitions

① Polynomial Regression is a way of modeling data by fitting independent and dependent variables to an nth degree polynomial.

② Quadratic Regression is used when the independent and dependent variables seem to have a quadratic relationship

B. Using My Calculator to Do Regression…

B. Examples① The table at the right shows the height of a

column of water as it drains from its container. Create a mathematical model that could be used to represent the water level over a period of time. Use your model to…

a. Estimate the water level at 35 seconds

b. Estimate the water level at 25 seconds

c. Use the model to predict what the water level will be after 3 minutes. Is this prediction reasonable? Explain!

Elapsed Time Water Level

0 s 120 mm

10 s 100 mm

20 s 83 mm

30 s 66 mm

40 s 50 mm

50 s 37 mm

60 s 28 mm

B. Examples① The table at the right shows the height of a

column of water as it drains from its container. Create a mathematical model that could be used to represent the water level over a period of time. Use your model to…

a. Estimate the water level at 35 seconds

b. Estimate the water level at 25 seconds

c. Use the model to predict what the water level will be after 3 minutes. Is this prediction reasonable? Explain!

Elapsed Time Water Level

0 s 120 mm

10 s 100 mm

20 s 83 mm

30 s 66 mm

40 s 50 mm

50 s 37 mm

60 s 28 mm

The model y=0.0092x2-2.1036+120.333 only works for for a range of 0 ≤ y ≤ 120mm and a domain of 0 ≤ x ≤ 108.57 secs. Where y is the level of the water in the tank in mm and x is the time gone by in seconds. The water level starts at 120mm (y-intercept) and drains until the water level is at y=0 (x-intercept). Thus, at 108.57 seconds there is no more water in the tank (x-intercept). When we are asked to predict where the water level in the tank will be after 3 minutes which is equivalent of 180 seconds we know the model we have won't work because 180 seconds is outside of the domain. Thus, the quadratic model we have is limited to the domain of 0 ≤ x ≤ 108.57 secs. All we can really say is after 3 minutes the water level is still going to be at y=0.

Homework

FLUENCY Pg. 246 (23-26, 31, 38, 42)

UNDERSTANDING Pg. 246 (34, 37, 39)

TAKE YOUR CALCULATOR HOME AND BRING IT BACK!!!Or Google Search “Quadratic Regression Calculator On-line” and use one of them

Homework

COMPLETE the stations check (19 Fluency problems 4 calculator problems)

MUST be completed for in class activity tomorrow

Suggestion: Time yourself! How long does it take you to answer all 23 questions? Should be under 50 minutes to finish during a class period

Finals are Here…• 1 (5) day and counting

• THREE PARTS1. Fluency (All Multiple Choice, must do all)2. Understanding (Three Create mathematical argument, must

do two)3. Model Card & Mind Map created in class*

• What Can I Do?– At home…use class website to prepare– Mind map using your notes– Annotate your own notes; use post-its to mark things you

have questions on

Three Finals…

• Yes You Can…– Use graphing calculator– Memorize Formulas

• No You Cannot…– Use Mind Map– Use Formula Sheet

5 Extra Credit Points…DUE FRIDAY!!!• Write a Thank You Letter

• Paid for the calculators we use as a class– Dear Kathy Mazza– Dear Sarah Kate– Dear Chris

• In the letter– Introduce your self (do not use your last name)– Tell them you are in Algebra II (no do say school or classroom

number) – Thank them for donating the calculator – Explain what you do with the calculators (graphing polynomials,

quadratic regression) and why they are a useful tool– Show an example/picture of that the calculator does– Thank them again & sign your FIRST NAME ONLY

Who Will Your Group Be?

• Maximum of 4• Minimum of 3

• List the people you are COMMITED to working on difficult problems with

Guess Who…

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