Name: Block:

Unit 5: Polynomial Functions

Day 1 Introduction, Factoring, Operations with Functions

Day 2 Compositions of Functions

Day 3 Quiz: Days 1 & 2 Graphing Polynomials; End Behavior; Zeros & Multiplicity

Day 4 Synthetic Substitution; Fundamental Theorem of Algebra Day 5 Applying the Remainder and Factor Theorem

Day 6 Finding All Rational Zeros

Day 7 Test Review

Day 8 Test

Tentative Schedule of Upcoming Classes

Day 1 B Thurs 1/14 Day 1 Notes: Introduction to Polynomials,

Factoring, Operations A Fri 1/15

Day 2 B Tues 1/19

Day 2 Notes: Composition of Functions A Wed 1/20

Day 3 B Thurs 1/21 Quiz: Days 1 & 2 Day 3 Notes: Graphing Sketching A Fri 1/22

Day 4 B Mon 1/25

Day 4 Notes: Synthetic Substitution A Tues 1/26

Day 5 B Wed 1/27 Day 5 Notes: Applying the Remainder & Factor Theorem A Thurs 1/28

Day 6 B Mon 2/1 Day 6 Notes: Finding All Rational Zeros A Tues 2/2

Day 7 B Wed 2/3 Unit 5 Review A Thurs 2/4

Day 8 B Fri 2/5 Test: Unit 5 A Mon 2/8

Absent?

See Ms. Huelsman AS SOON AS POSSIBLE to get work and any help you need.

Notes are always posted online on the calendar. (If links are not cooperative, try changing to “list” mode)

Handouts and homework keys are posted under assignments

You may also email Ms. Huelsman at Kelsey.huelsman@lcps.org with any questions!

____

Need Help?

Ms. Huelsman and Mu Alpha Theta are available to help Monday, Tuesday, Thursday, and Friday mornings in L506 starting at 8:10.

Ms. Huelsman is in L402 on Wednesday mornings.

Need to make up a test/quiz?

Math Make Up Room schedule is posted around the math hallway & in Ms. Huelsman’s classroom

Day 1 Notes: Introducing…POLYNOMIAL FUNCTIONS

In these notes we will review factoring & extend our knowledge to sums and differences of cubes, define operations on polynomials: +, -, x quadratics, cubics and more! So that we can combine any polynomials and find their zeros. FACTORING GUIDE: 1. Look for a GCF - If we have one, factor it out but KEEP IT as a factor. 2. Is what remains a BINOMIAL? - Is it a difference of TWO SQUARES? - Is it the SUM OR DIFFERENCE OF TWO CUBES? 3. Is what remains a TRINOMIAL? - Try to factor…if the leading coefficient isn’t one, remember: factor by grouping! FACTOR the following COMPLETELY. Be sure to look for a GCF if a≠1 or if there are multiple x’s! 1. 2x3 – 8x 2. 4x2 – 64 3. 4x2 – 81 4. 5x2 + 15x + 10 5. 10x2 – 25x – 15

ANOTHER PATTERN: SUM and DIFFERENCE of PERFECT CUBES Remember our difference of two squares? 25x2 – 49 x4 – y4

What about cubes? x3 – 8 or x3 + 8 ? Factoring Sums or Differences of Two CUBES a3 + b3 = (a + b)(a2 – ab + b2) [SOAP the signs…. Same, Opposite, Always Positive]

a3 – b3 = (a – b)(a2 + ab + b2) 1. x3 – 8 2. 2x3 + 250 (hint: GCF!) 3. 1000x3 + 1 4. y3 – 64x3 Discuss: What is the difference between (x3 – 8) and (x – 8)3

A polynomial function is a function of the form: f(x) = anxn + an-1xn-1 + . . . + a1x + a0

Note: all coefficients must be real numbers and all exponents of variables must be whole numbers!!

an is the ______________________, n is the ___________, and a0 is the _______________ A polynomial function is in STANDARD FORM if its terms are written in descending order of exponents from left to right. Example: f(x) = 2x3 – 5x2 – 4x + 7 leading coefficient_____, constant_____, and degree_____. Example: Determine whether the function is a polynomial function. If so, write it in Standard Form and state its degree, type, and leading coefficient. 1. f(x) = 6x1/2 – 5x 2. g(x) = -4x2 – 8x5 + 10 + x4 Exponents whole numbers? ____ Exponents whole numbers? ____ Coefficients real numbers? ____ Coefficients real numbers? ____ Polynomial / not a polynomial Polynomial / not a polynomial Standard Form: Standard Form:

Degree: Degree:

Type: Type:

Leading Coefficient: Leading Coefficient:

3. Write a polynomial function of degree 7 with leading coefficient of 2

Operations with Polynomials – Adding, Subtracting, Multiplying

1. (2x2 - 8) + (x3 + 4x2 - 12x + 4)

2. (2x2 - 8) – (x3 + 4x2 - 12x + 4)

What does it mean to square something? 3. (3x2 + 7)2

4. (a – 6)(2a + 5)(a + 1)

6. (x – 3)(x2 + 3x + 9) Let’s review transformations of cubic functions from the video homework

1. y = (x + 1)3 – 2 2. y = – (x – 2)3 +1

The “vertex” (CENTER) is ________ . The “vertex” (CENTER) is ________ .

Summary of Transformations of Cubic Functions:

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

Day 2 Notes: Composition of Functions

In these notes we will review operations and apply this to find compositions of functions.

Warm up: Review of Day 1 – Operations & Factoring Polynomials 1. (2z2 – 1)(z + 5) = 2. (2z2 – 1) – (z + 5) What is the difference between these two questions? 3. Factor completely: a.) x3 + 8 b.) 125x3 – 64 c.) 343x3 – 216 d.) 64x3 – 27 4. Decide whether each function is a polynomial. If so, rewrite in standard form, state the degree & type, and the leading coefficient.

Function Standard form of the

polynomial (if yes)

Degree of

polynomial

Type of

Polynomial

Leading

Coefficient 12 1y x= +

( ) 2xf x =

g(x) = - 4x2

p(x) = -3x2 – x3 – 3+ x

y = 1 - 5x

13y x−=

f(x) = –5x2 + 7+ 2x3– 4x

Another operation with Polynomials – Composition of Functions **Composing functions is a fancy way of embedding one function into another. When you consider the DOMAIN of a composition function, f(g(x)), you have to look at the restrictions for the input AND the result Let f(x) = x + 7 and g(x) = 2x + 3 Let’s evaluate… f(1) g(1) What does this mean? What does this mean? f(g(1)) g(f(1)) What does this mean? What does this mean?

What is the domain of f(x)? What is the domain of g(x)? Sketch each graph to get a better idea. Now let’s create a WHOLE NEW function Find f(g(x)) using the functions above. Find g(f(x)) using the functions above. Let’s find f(g(1)) using that function Let’s find g(f(1)) using that function How does this compare to f(g(1)) above? How does this compare to f(g(1)) above? What is the domain of f(g(x))? What is the domain of g(f(x))? Sketch each graph to get a better idea.

Let’s try another composition of functions… Let f(x) = x2 + 2 and g(x) = x + 4

a. Evaluate f(g(-5)) b. Evaluate g(f(-5))

Find f o g = f(g(x)) Find g o f = g(f(x))

Let’s check f(g(-5)) = Let’s check g(f(-5)) = Which way do you prefer finding f(g(5))? What is the domain of f(x)? What is the domain of g(x)? Sketch each graph to get a better idea. What is the domain of f(g(x))? What is the domain of g(f(x))? Sketch each graph to get a better idea.

Day 3 Notes: Graph Sketching In these notes we will apply prior knowledge of end behavior and transformations and apply

this to sketch polynomials.

Part 1: Graphing with a focus on End Behavior Sketch the general shape of each function. Use your calculator!

f(x) = 2x f(x) = x + 3 f(x) = x3 f(x) = 2x3 + 1

Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______ f(x) = -2x f(x) = -x + 3 f(x) = -x3 f(x) = -2x3 + 1

Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______ f(x) = 2x2 f(x) = x2 + 2x + 3 f(x) = x4 f(x) = x4 +2x3 - 1

Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______ f(x) = -2x2 f(x) = -x2 + 3 f(x) = -x4 f(x) = -x4 +3x3 - x+ 1

Is the leading coefficient positive or negative? ______ Is the degree even or odd? _______ What do you notice with the graphs: An odd degree function with a positive leading coefficient:

An odd degree function with a negative leading coefficient:

An even degree function with a positive leading coefficient:

An even degree function with a negative leading coefficient:

We can be more formal with these descriptions. Reminder about end behavior patterns: Describe the degree (even or odd) and leading coefficient (positive or negative) of the polynomial function. Then describe the end behavior of the graph of the polynomial function. 1. 2. Degree _______ (even / odd) Degree _______ Leading Coefficient __________ (+ / -) Leading Coefficient __________

( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

3. y = -3x5 – 6x2 + 3x – 8 4. h(x) = 6x8 – 7x5 + 4x Degree _______ Degree _______ Leading Coefficient __________ Leading Coefficient __________ Rough sketch based on what we know:

( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

One last idea: Zeros and Multiplicity

Example: Polynomial function in FACTORED form f(x) = (x + 4)2 (x + 1) (x – 3) Degree: ___________ Leading Coefficient: _________ Zeros and multiplicity:

What can we tell about a polynomial function by looking at its equation in factored form? When a factor (x – k) of a function f is raised to an ODD power,

the graph of f crosses the x-axis at x = k.

When a factor (x – k) of a function f is raised to an EVEN power, the graph of f is tangent to the x-axis at x = k. (It “bounces off” the x-axis)

#1 For the function f(x) = 3x5 (x – 5) (x + 1)2 (x + 2)3,

What is the multiplicity at: Does the function bounce or cross at:

a. x= 0 a. x= 0

b. x = 5 b. x = 5

c. x = –1 c. x = –1

d. x = –2 d. x = –2

2. y = 2 (x – 1)2 (x + 2) Degree: ________ Leading Coefficient: ________

End behavior: ( ) _____( ) _____

As x then f xAs x then f x

→ −∞ →→ +∞ →

Zeros and multiplicity: _______ _______ _______ Double root(s): ____________________

Graph Sketch

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

Practice of Zeros and Multiplicity

Polynomial function in FACTORED form f(x) = -2x (x -5)(x + 1) Degree: ___________ LC: _________ Zeros and multiplicity: Double Root(s)? YES or NO If yes, Name them ________________

End behavior: ( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

REMEMBER:

• When a factor (x – k) is raised to an ODD power, the graph CROSSES the x-axis at x = k.

• When a factor (x – k) is raised to an EVEN power, the graph BOUNCES/TURNS on the x-axis at x = k.

1. y = –x (x – 2) (x + 3)2 Degree: ________ Leading Coefficient: ________

End behavior: ( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

Zeros and multiplicity: _____________________________ Double root(s)? YES or NO If yes, Name them _______________

Graph Sketch

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

2. y = (x – 5)2 (x -3) Degree: ________ Leading Coefficient: ________

End behavior: ( ) _____( ) _____

As x then f xAs x then f x

→ +∞ →→ −∞ →

Zeros and multiplicity: _______________________ Double root(s): _____________________

Graph Sketch

3. Sketch the polynomial function described. Degree = 4 Leading Coefficient = 1 Zeros: –2 (multiplicity of 1) 4 (multiplicity of 2) 7 (multiplicity of 1)

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

Day 4 Notes: Synthetic Substitution & “Fundamental Theorem of Algebra” In these notes we will learn how to evaluate a polynomial function by synthetic substitution and introduce the idea of the fundamental theorem of algebra so we can better predict the behavior of a polynomial’s graph. We just finished learning about END BEHAVIOR (as x ). Well, what about behavior in the "middle"?

You know how to graph this…. f(x) = x2 – 2x – 8 What is the function value at x= 0? How about x= -1? Evaluating Polynomial Functions – Why do we want to do this? What does it mean to evaluate? Evaluate by Direct Substitution (Method 1) Use Direct Substitution to evaluate f(x) = -3x3 + x2 – 12x – 5 when x = -2 In other words…”find f(-2)” Evaluate by Synthetic Substitution (Method 2) Use Synthetic Substitution to evaluate f(x) = -3x3 + x2 – 12x – 5 when x = -2 (same problem as above) In other words…”Find f(-2)” (Note this polynomial is in STANDARD FORM!) Multiply -2 -3 1 -12 -5 Add

1. Bring Down 2. Multiply and Add 3. Multiply and Add 4. Multiply and Add…

x

y

SO……f( )= Use Synthetic Substitution to evaluate f(x) = -2x4 – x3 + 4x – 5 when x = -1 In other words… **Write the coefficients of f(x) in order of descending exponents (there is 0x2) Multiply -1 -2 -1 0 4 -5 Add SO……f( )= Use Synthetic Substitution to evaluate f(x) = -2x4 + 3x3 - 8x + 13 when x =2 In other words, SO……f( )= Use Synthetic Substitution to evaluate f(x) = x3 - 27 when x = 3 In other words, SO……f( )= SO, synthetic substitution can also be used to find _____________ of a function

41. Use direct substitution given x = 2. Write as an ordered pair: ____________ f(x) = 5x3 – 8x2 + 5x – 1 Write in function notation: ___________ Is the ordered pair a zero? ___________ 42. Use synthetic substitution given x = 5. Write as an ordered pair: ____________ Write in function notation: ___________ Is the ordered pair a zero? ___________ 43. Use synthetic substitution given x = –2. Write as an ordered pair: ____________

Write in function notation: ___________

Is the ordered pair a zero? ____________

Direct and Synthetic Substitution

Direct Substitution

1. Substitute the x-value into each x in the function.

2. Make sure to put parentheses around the x-value.

3. Simplify.

* Recall, −3( )2 = 9

−32 = −9

Synthetic Substitution

1. Put the x-value on the outside of the “L”

2. Put the coefficients on the inside. Make sure to have “0” as placeholders if the function is missing any terms.

3. Bring down the first coefficient. 4. Multiply and Add. Repeat. 5. You can write the solution as an ordered

pair (x, y) or as a function f(x) = y. *If the final sum is zero, then the x-value is a zero/x-intercept/solution/root!

IF we now have a quick way to FIND AND GRAPH POINTS of a polynomial and TEST FOR ZEROS,

how do we GRAPH THE REST?

The Fundamental Theorem of Algebra An nth degree polynomial function has exactly ____ solutions.

1. How many solutions does the equation x4 + 8x2 – 2x + 2 = 0 have? ______ How many turning points does it have?

2. How many zeros does the function g(x) = x3 – x2 – 3x – 3 have? _____

How many turning points does it have?

However, some of those zeros may be imaginary! If a quartic function has 4 total solutions, let’s find combinations of real & imaginary solutions:

Complex and Irrational Conjugates – Imaginary (i) and irrational ( ) solutions always come in pairs.

Complex Conjugate Theorem: if is a zero, then is also a zero

Irrational Conjugates Theorem: if is a zero, then is also a zero Discussion: Write out the quadratic formula. WHY are these theorems TRUE?

Given a zero, identify another zero (its pair!) 1. _________ 2. ________ 3. _______ 4. ___________

Given Factors, Identify the Zeros 1. (x – 5)(x – 2) _________ 2. (x + 6)2 _________ 3. 3x(x – 2) _________ 4. 4x3(2x – 1) _______ 5. (3x – 1)(6x + 5) __________

bia + bia −

ba + ba −

5− i26 − 723− i3

Given a List of Zeros, Write as a List of Factors (working backwards!) 1. 7 __________ 2. x = -2 __________ 3. x = 5, -8 __________ 4. x = __________________

5. x = 32± __________________ 6. x = ___________

7. x = _______________ 8. x = _________________

Write a Polynomial Function given the Zeros 1. Find the polynomial with a leading coefficient of 1 that has the given zeros: 4, 1, -2

First, what degree will this be? (hint: power = number of zeros!) Second, End behavior: As , as ,

Write f(x) in factored form: Change to Standard Form: 2. Find the polynomial with a leading coefficient of 1 that has the given zeros: 5, 3, Identify all zeros: _________________________

What degree will this be? End behavior: As , as ,

Write f(x) in factored form: Change to Standard Form:

6±

32

37

− 5,41

−

x →+∞ _____)( →xf x →−∞ _____)( →xf

i2−

x →+∞ _____)( →xf x →−∞ _____)( →xf

Review - What We Know: Graphs of Polynomials Leading coefficient > 0 , Odd Degree

Leading coefficient < 0 , Odd Degree

Leading coefficient > 0 , Even Degree

Leading coefficient < 0 , Even Degree

As x + , f(x) As x - , f(x)

As x + , f(x) As x - , f(x)

As x + , f(x) As x - , f(x)

As x + , f(x) As x - , f(x)

Let’s review some vocabulary associated with the graphs: Turning points Relative Max / Relative Min Absolute Max / Absolute Min

Graph 3x4 + x3 - 10x2 + 2x + 7 using your calculator. Sketch its graph below.

a. Determine the number of zeros for the polynomial _________________

b. Determine the number of real zeros for the polynomial ______________

c. Determine the number of turning points _________________________

d. Does the graph have relative minimums or maximums? ______________

e. Does the graph have absolute minimums or maximums? ______________

f. Describe the end behavior of the graph: As , as ,

∞

∞

∞

∞

∞

∞

∞

∞

x →+∞ _____)( →xf x →−∞ _____)( →xf

Day 5 Notes: Applying the Remainder & Factor Theorem In these notes we will learn how to find zeros using division to help sketch a polynomial’s graph.

Dividing Polynomials: There are 2 ways to divide polynomials - long division and synthetic division. Synthetic Division: Divide x 3 962 2 −−+ xx by (x – 2)

1. First, the divisor must be in the form: (x – k) What is k in the divisor (x - 2)? 2. Write down the coefficients of each term. If a term is missing, you must use a zero as a place-holder. 3. Then drop, multiply and add, multiply and add, etc.

4. The answer is one power lower than the original (a “depressed polynomial”). The last term is the remainder.

Let’s verify by looking at long division. You try: Divide f(x) = x4 + 4x3 + 16x – 35 by (x + 5) using SYNTHETIC division.

1. First, the divisor must be in the form: (x – k) What is k in the divisor (x - 2)? 2. Write down the coefficients of each term. If a term is missing, you must use a zero as a place-holder. 3. Then drop, multiply and add, multiply and add, etc.

4. The answer is one power lower than the original (a “depressed polynomial”). The last term is the remainder.

Remainder and Factor Theorems

Basically, the remainder = the function value (what we get when we do synthetic substitution!)

As long as we get 0 as the remainder / function value, we can write the zero as a factor.

Application #1: Factoring a Polynomial

Example 1: Factor f(x) = x3 – 2x2 – 40x – 64 completely given that x – 8 is a factor. Step 1. Synthetic Division - because (x – 8) is a factor, f(8) = 0, and x = 8 is a solution. Step 2. Write answer as a product of 2 factors. Step 3. List ALL factors. So… Factored form of f(x) = x3 – 2x2 – 40x – 64 = ( ) ( ) ( ) Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial, f(x) = x3 – 2x2 – 40x – 64

Example 2: Factor f(x) = 3x3 – 4x2 – 28x – 16 completely given that x + 2 is a factor. Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial, f(x) = 3x3 – 4x2 – 28x – 16.

Application #2: Finding the Zeros, Roots, or Solutions Example 1: One zero of f(x) = x3 + x2 – 16x – 16 is 4. Find all the zeros. 1. Synthetic Division – because f(4) = 0, x = 4 is a zero/solution, and (x – 4) is a factor What do we use for synthetic division, 4 or - 4? 2. Write the “depressed polynomial” as a product of 2 factors. Then find the zeros. Remember: you know multiple methods to find the zeros/solutions of a quadratic function! Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial, f(x) = x3 + x2 – 16x – 16

Example2: One zero of f(x) = x3 + x2 + 2x + 24 is -3. Find all the zeros. Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial f(x) = x3 + x2 + 2x + 24

Challenge: We can even do this with higher power polynomials! One zero of f(x) = x4 + 3x2 - 4 is 1. Find all the zeros. (Hint: how many should there be?!) Now use your knowledge of end behavior and solutions to sketch a graph of the given polynomial, f(x) = x4 + 3x2 - 4.

Day 6 Notes: Finding All Rational Zeros In these notes we will learn how to find ALL zeros without being given a factor or zero to start! Rational Zero Theorem gives you possible zeros in case you do not have access to the graphing calculator.

LUCKILY…you will use your calculator to graph the polynomial function, using the list of possible rational zeros as a guide. Using the graph you will identify a zero to “test” using synthetic division. (Should your remainder be 0?) Example 1: Find all zeros of f(x) = x3 + 2x2 - 11x – 12 1. Using The Rational Zero Theorem, list all possible rational zeros: 2. Graph the polynomial function in the calculator and estimate the real zeros. Do any match your list? 3. Test one of your zeros (based on the graph) using synthetic division. Remember…it is a solution if the remainder (or output) = 0.

4. Write as a new polynomial (one power lower), then find the remaining zeros (Ways we know to solve: ______________________, ______________________, ______________________)

5. List ALL zeros:

Example 2: Find all zeros of f(x) = x3 - 8x2 + 11x + 20 1. Using The Rational Zero Theorem, list all possible rational zeros: 2. Graph the polynomial function in the calculator and estimate the real zeros. Do any match your list? 3. Test one of your zeros (based on the graph) using synthetic division. Remember…it is a solution if the remainder (or output) = 0.

4. Write as a new polynomial (one power lower), then find the remaining zeros (by factoring, using quadratic formula, or completing the square).

5. List ALL zeros: Example 3: Find all zeros of f(x) = x3 + 3x2 - 4 (follow all steps used above)

Sketch a graph of your function:

Example 4: Find all zeros of f(x) = x3 + x2 - 3x - 6

Sketch a graph of your function:

Example 5: Find all zeros of f(x) = 4x3 - 12x2 - x + 15

Sketch a graph of your function:

Unit Review: Polynomial Operations 1. (x3 + 1) + (3x2 + 5) + (9x2 – 2x) 2. (x2 – 5x + 1) – (3x2 + x – 5)

3. (2x + 7)2 4. 2 (x – 4)3

Factor sum/difference of perfect cubes. 5. x3 + 27 6. 8x3 – 1

7. 2x3 + 16 8. 64x3 – 125y3

9. f (x )= 2x(x +1)(x −1)3

Polynomial: Yes or No Form: Standard or Factored 10. f (x )= 2x +3x − 5 Polynomial: Yes or No Form: Standard or Factored 11. Put the polynomial function into standard form: f (x )= 2x 3 − x 5 +10− x Given the factors, find the zeros. Given the zeros, find the factors. 12. Factor: (x + 3) Zero: ________ 15. Zero: 9 Factor: __________ 13. Factor: (x) Zero: ________ 16. Zero: –3i Factor: ___________

14. Factor: (5x – 1) Zero: ________ 17. Zero: Factor: ___________

35

Given a zero, find another. 18. Zero: 5i, ______________ 19. Zero: 4 – 7i, ______________

20. Zero: − 17 , ______________ 21. Zero: 5− 2 3 , ______________ Fill out the chart about end behavior.

Function Degree LC Rough Sketch

(with maximum turns)

End Behavior

22 f (x )=−4x3+2x +1

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

23 f (x )= 2(x +5)(x −2)

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

24 f (x )= x(x −3)(x +5)

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

25 f (x )= 5x(x +3)2(x +1)3

x →∞, f(x)→ _____ x →−∞, f(x)→ _____

26. Find the polynomial with a leading coefficient of 2 that has the given zeros: 1, –2i Write f(x) in factored form: ________________________________________ Change to Standard Form:

27. Given f(x) = x2 + 1 and g(x) = x – 5 a.) Find f(g(x)) and write in standard form b.) find g(f(x)) and write in standard form c.) Find f(g(2)) d.) find g(f(2))

27. y = x (x + 4)3 (x – 2)2 Degree: ________ Leading Coefficient: ________ End behavior: Zeros and multiplicity: ____________ ____________ ____________

Sketch:

28. y = –3 (x – 1)2 (x + 4)3 Degree: ________ Leading Coefficient: ________ End behavior: Zeros and multiplicity: ____________ ____________ ____________

Sketch

29. Graph y= (x - 2)3 + 3 Center: ____________ 30. Graph y= – (x – 2)3 +1 Center: ____________

x →∞, f(x)→ _____

x →−∞, f(x)→ _____

x →∞, f(x)→ _____

x →−∞, f(x)→ _____

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y

−9 −6 −3 3 6 9

−9

−6

−3

3

6

9

x

y