Unit I - Chapter 3 Polynomial Functions 3.1 Characteristics of ......Math 3200 Unit I Ch 3 - Polynomial Functions 2 Leading Coefficient is the number in front of the term with the

Math 3200 Unit I Ch 3 - Polynomial Functions 1

Unit I - Chapter 3 Polynomial Functions

3.1 Characteristics of Polynomial Functions

(I) Identifying Polynomial Functions and Graphs of Polynomial

Functions

Terminology

End Behavior of a polynomial function graphically refers to what

is happening to the function

as x approaches +∞ or –∞

Degree of a polynomial function refers to the highest exponent on

a variable

Ex. f(x) = 7 + 3x2 – 2x

4

x

y

End Behavior

Curve extends up into quadrant 1 (as x

approaches +∞) and down into quadrant

3 (as x approaches –∞).

The Degree is __________

Goal: To Understand some Basic Features of Polynomial functions:

Continuous

Max. turns: n – 1

Leading coefficient effect/end behavior

Even vs odd degree

Comparing f(x) = x4 “flatter” than f(x) = x

2

Identifying Polynomial Functions


Leading Coefficient is the number in front of the term with the

highest exponent

Ex. f(x) = x5 – 7x

3

Identifying Polynomial Functions

The Leading Coefficient is __________


1. Use graphing technology (using graphing software.

https://www.desmos.com/) to graph each function and complete the table.

Function

Shape

End

Behaviour Degree # of Turns

Leading

Coefficient

Number of

x-Intercepts

Max/Min?

y-int.

y = x + 2

y = –3x+1

y = x2 – 4

y = –2x2– 2x+4

y = x3– 4x

y = –x3+3x–2

y = 2x3 + 16

https://www.desmos.com/


2. What does the degree indicate about the behavior of the graph to the left

and right?

Function

Shape

End

Behaviour Degree # of Turns

Leading

Coefficient

Number of

x-Intercepts

Max/Min?

y-int.

y = –x3 – 4x

y = –x4+x

3+4x

2– 4x

y = x4 + 2x

2 + 1

y = x5 – 1

y = x5–2x

4 –3x

3+5x

2

+ 4x – 1

y = –x5+ x

4

+8x3+8x

2 – 16x – 16

y = x(x + 1)2(x + 4)

2


3. How is the sign of the leading coefficient and the end behavior related?

4. Can you predict the number of turns from the equation? Why or why not?

5. Which feature of the equation relates to the number of x-intercepts?

6. Does every function have either a maximum or minimum? Why or why not?

7. How is the y-intercept determined?

(P.114 – 115 #1 - #5, #13)


3.1 Continued

Analyzing Polynomial Functions Algebraically and Graphically

(I) Rules for graphing odd or even polynomial functions

On the same graph sketch each pair of polynomial functions.

(A) f(x) = x2 and g(x) = x

4 (B) h(x) = x

3 and p(x) = x

5

What end behavior is exhibited when leading coefficients are negative for even

and odd functions?

x- 3 - 2 - 1 1 2 3

y

- 4

- 3

- 2

- 1

1

2

3

4

x- 2 - 1 1 2

y

- 4

- 3

- 2

- 1

1

2

3

4

Even Degree Functions f(x) = axn

With positive leading coefficients

exhibit behavior in the ________

and ____________ quadrants

Graphically there is a ___________

effect when the value of increases

over the interval

_______________.

Odd Degree Functions f(x) = axn




Graphically there is a ___________

effect when the value of increases

over the interval

_______________.


(II) Graph each of the given functions and answer the indicated characteristics.

Function

Graph End Behavior

Degree

(Odd/Even)

# of

x-Intercepts

Constant

Term

y-int.

f(x) = –2x + 1

f(x) = –x2 – 2x + 3

f(x) = –x3 – 2x

2 + x + 2

f(x) = –x4–5x

3–5x

2+5x+ 6

f(x) = –x5+5x

3–4x

x

y

x

y

x

y

x

y

x

y


Summary of Characteristics of Polynomial Functions

(A) Degree of Polynomial Function

(B) Constant Term of a Polynomial Function

(C) The Number of Real x-intercepts

Odd Degree Functions f(x) = axn

With negative leading coefficients






Even Degree Functions f(x) = axn

With negative leading coefficients






For odd/even functions, the ___________________ corresponds to the constant term.

Odd Degree Functions f(x) = axn + …. + c

At least one x-intercept to a

maximum of ___ intercepts

No max or min points

Domain _______ Range ________

Even Degree Functions f(x) = axn + …. + c

Zero x-intercepts to a maximum of

___ intercepts

Max or min point depends on

direction

Domain _______

Range depends on


(III) Review Questions:

(a) Identify the features of the graph related to the function

f(x) = −3x2 + 9x + x

5.

Leading Coefficient ________

Degree______

End Behavior _________________________________________

y-intercept____________

Number of possible x-intercepts_____________________

Max or min values? _____________

(b) Match the functions with the appropriate graphs.


(c) How many turns can the graph of a polynomial function of degree 5 have?

Explain.

____________________________________________________________

____________________________________________________________

(d)







The Factor Theorem (Part I)

Determining the Remainder for a Factor of a Polynomial

Example: Determine the remainder when (x3 + 3x

2 - 4) is divided by (x + 2).

Conclusion: Since P( ) = ___ (x + 2) is a _________

Factors (x - a) of a Polynomial Expression and Zeros of a

Polynomial Function

(I) Attaining Linear Factors (x - a) of a Polynomial Expression

Example: If (x + 2) is a factor of (x3 + 3x

2 - 4) then determine all other linear

factors.


(II) Attaining Zeros of a Polynomial Function

Example: If a polynomial function P(x) = x3 + 3x

2 - 4

expressed in factored form as P(x) = (x + 2)2 ( x - 1)

then determine the zeros.

Linear Factors and Zeros of a Function

The Factor Theorem

(x - a) is a factor of P(x) if and only if __________

Example: Verify if x + 3 is a factor of P(x) = 2x3 + x

2 - 13x + 6.


Integral Zero Theorem

How can we determine all of the factors of a polynomial function?

Example: If x + 1 is a factor of P(x) = x3 - 4x

2 + x + 6 then determine

all other factors.

Which term in the polynomial function P(x) = x3 - 4x

2 + x + 6 has

factors ____, ____ and _____ ?


Example: Fully factor the given the polynomial

function P(x) = x3 + 3x

2 - 6x – 8

(i) List all possible integral zeros.

(Apply Integral Zero Theorem)

(ii) Verify one factor. (Apply Factor Theorem)

(iii) Reduce polynomial to a lesser degree.

(Apply Synthetic Division)

(iv) Express P(x) in factored form


Factoring Higher Degree Polynomials

(I) Factoring Cubic (degree 3 with 4 terms) Polynomials by Grouping

Example: Factor fully.

(a) x3 + 6x

2 - 4x - 24 (b) x

3 - 4x

2 - 9x + 36

(II) Factoring Quartics (degree 4) Polynomials

Example: Factor fully. x4 - 5x

2 - 36

Text Questions: #1b,c #2a,c #3b,d #4a,c #5b,c,d #6a,c,d #7b,c


The Factor Theorem (Part II)

Factoring polynomials P(x), that contain non - integer zeros by applying IZT,

FT and SD

Modelling and solving problems involving polynomial functions

(I) Factoring a Polynomial P(x) that also contains non - integer zeros

Example: Given the polynomial function P(x) = 6x3 + 5x

2 - 2x – 1

(a) Use the Integral Zero Theorem to list all possible integral factors

(b) Verify one of the factors using the factor theorem

(c) Apply synthetic division to determine the remaining factors

(d) Express P(x) = 6x3 + 5x

2 - 2x - 1 in factored form

Factored Form P(x) =


Note: The linear factors will now produce ____________ zeros since

the ___________________ is no longer 1.

Summary - To solve polynomial functions:

List all possible integral zeros

Use the factor theorem to verify a zero

Use synthetic division to reduce the polynomial

Repeat the above process to determine the remaining factors or the

polynomial is reduced to a trinomial that can factor.

Example: Fully factor the polynomial function f(x) = 4x3 - 12x

2 + 5x + 6


(II) Modelling and Solving Problems involving Polynomial Functions

Example: P.134 of Textbook

P.134 - 135 #9, #11, #13, #16, C1


Section 3.4 Equations and Graphs of Polynomial Functions

(Part I)

Investigating the relationship between zeros, x - intercepts and roots

Sketching the graph of polynomial functions

Modelling and solving problems involving polynomial functions

(I) Investigating the relationship between zeros, x - intercepts and roots

Example: Given the polynomial function f(x) = x4 + 2x

3 - 4x

2 - 2x + 3

(a) Use graphing technology to sketch the graph and determine

the x - intercepts from the graph.

x - intercepts of the graph are:


Attaining the Zeros from a Function

(b) Factor the polynomial function f(x) = x4 + 2x

3 - 4x

2 - 2x + 3 then use

the factors to determine the zeros.

(c) Solve the equation x4 + 2x

3 - 4x

2 - 2x + 3 = 0 to determine the roots.

What do you notice about the x - intercepts of the graph, zeros of the function

and roots of the equation?


(II) Investigating the Graphs of Polynomial Functions and the

Multiplicity of a Zero

For each Polynomial Function:

State the x – intercepts

Use technology to sketch the graph

State the multiplicity of each zero

Indicate the intervals where the function is positive

(above the x – axis) or negative (below the x – axis)

Multiplicity of a Zero

The number of times the zero of a

polynomial occurs

The shape of a graph at a zero depends on

the multiplicity


Function

x-intercepts

Graphs/Multiplicity of

zeros

Intervals

1. f(x)=(x+1)(x-1)(x+2)

2. f(x)=(x-1)2(x+2)

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345


Function

x-intercepts

Graphs/Multiplicity

of zeros

Intervals

3. f(x)=(x-1)3

4. f(x)= x2

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345


Function

x-

intercepts

Graphs/Multiplicity of

zeros

Intervals

5. f(x)=x3

6. f(x)=x4

7. f(x)=–(x+1)3(x – 2)

2

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345

x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5

y

- 5- 4- 3- 2- 1

12345


NOTE: Multiplicity of a zero and characteristics of the graph

(I) Zero of multiplicity One (II) Zero of multiplicity Two

(III) Zero of multiplicity Three

x

y

x

y

x

y

Odd/Even Multiplicity

(i) The graph of a polynomial function

_____________ the x-axis where the intercepts

have odd multiplicity.

(ii) The graph of a polynomial function

_____________ the x-axis where the intercepts

have even multiplicity.


(III) Sketching the Graph of a Polynomial Function

Examples: For each of the given the polynomial functions

(i) determine the degree, leading coefficient, end behaviour,

zeros/x - intercepts, y - intercept and interval where the

function is positive or negative.

(ii) use the information above to sketch the graph.

(a) f(x) = -(x + 2)2(x - 1)

2

(b) y = x(x - 2)3


(c) y = -2x3 + 6x - 4

(IV) Determining the Equation of a Polynomial Function from a Graph

Examples: Use the graph to determine the equation of the given

polynomial function.

(a)


(b)

P. 147 - P.149 #1a, #2c, #3, #4b,c #7a, b, c #8, #9c, d, e #10a, c

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Unit I - Chapter 3 Polynomial Functions 3.1 Characteristics of ......Math 3200 Unit I Ch 3 - Polynomial Functions 2 Leading Coefficient is the number in front of the term with the