polynomial functions day 1 and 2. polynomial functions do now: find the range of yesterday’s exit...

Polynomial FunctionsDay 1 and 2

Polynomial Functions

• Do now: Find the

Range of yesterday’s exit ticket problem!

• Exit Ticket: Start homework:

• Do not lose handout-

hmwk #5

• Objectives: Given a polynomial (many number) function, determine from the graph what degree is

• Find the “zeros” from the graph or the equation, in order to recognize equations of same degree

Polynomial Functions (day 2)

• Do now: Write the

TWO fundamental Rules of Algebra you (memorized?!)

• learned from yesterday’s powerpoint….!

• Exit Ticket:

• Sketch an exponential GROWTH function



Just a few definitions Polynomial function

a function with one or more terms

Ex) 2x5 – 5x3 – 10x + 9, because it has 4 terms.Ex) 7x4 + 6x2 + x has 3 terms

Degree the highest exponent power that is in a term

Ex) 5 x3 has a degree of “3”Ex) 10x6 has a degree of “6”

Highest Degree Highest degree that is in a polynomial. When you are asked for the degree of a polynomial, you are being asked for the highest degree.

Ex) 2x5 – 5x3 – 10x + 9 has a highest degree of 5Ex) 7x4 + 6x2 + x has a highest degree of 4


The largest exponent within the polynomial determines the degree of the polynomial.

Polynomial Function in

General Form

Degree Name of Function

1 Linear

2 Quadratic

3 Cubic

4 Quarticedxcxbxaxy 234

dcxbxaxy 23

cbxaxy 2

baxy

Fundamental Theorem of Algebra:

• Degree of the polynomial is the same as the number of “ups” and “downs” of its graph…

• Try the examples in notes.

Leading Coefficient

The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.

For example, the cubic function f(x) = -2x3 + x2 – 5x – 10 has a leading

coefficient of -2. This will play an important role in it’s graph…

2nd Fundamental Theorem of Algebra:

The number of zeros that a polynomial function has is equal to that function’s degree.

Explore Polynomials

Linear Function

Quadratic Function

Cubic Function

Quartic Function

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10

-60-55-50-45-40-35-30-25-20-15-10-5

510

Cubic PolynomialsLet’s look at the two graphs and let’s discuss the questions below.

1. How can you check to see if both graphs are functions?

3. What is the end behaviour for each graph?

4. Which graph do you think has a positive leading coeffient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph B

Graph A

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

Cubic PolynomialsEquationEquation

Factored form & Factored form & Standard formStandard form

X-InterceptsX-Intercepts Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

Factoredy=(x+1)(x+4)(x-2)

Standardy=x3+3x2-6x-8

-4, -1, 2 Positive

As x, y and

x-,

y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x+1)(x+4)(x-2)

Standardy=-x3-3x2+6x+8

-4, -1, 2 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}

The following chart shows the properties of the graphs on the left.

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

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

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

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12







Factoredy=(x+3)2(x-1)

Standardy=x3+5x2+3x-9

-3, 1 Positive

As x, y and

x-,

y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x+3)2(x-1)

Standardy=-x3-5x2-3x+9

-3, 1 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}


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

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

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

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12







Factoredy=(x-2)3

Standardy=x3-6x2+12x-8

2 Positive

As x, y and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R}

Factoredy=-(x-2)3

Standardy=-x3+6x2-12x+8

2 Negative

As x, y- and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R}


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

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

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

-12

-10

-8

-6

-4

-2

2

4

6

8

10

12

Quartic PolynomialsLook at the two graphs and discuss the questions given below.

1. How can you check to see if both graphs are functions?

3. What is the end behaviour for each graph?

4. Which graph do you think has a positive leading coeffient? Why?

5. Which graph do you think has a negative leading coefficient? Why?

2. How many x-intercepts do graphs A & B have?

Graph BGraph A

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

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

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

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

Quartic Polynomials

EquationEquation

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading




Factoredy=(x-3)(x-1)(x+1)(x+2)

Standardy=x4-x3-7x2+x+6

-2,-1,1,3 Positive

As x, y and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -12.95}

Factoredy=-(x-3)(x-1)(x+1)(x+2)

Standardy=-x4+x3+7x2-x-6

-2,-1,1,3 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 12.95}


-10 -8 -6 -4 -2 2 4 6 8 10

-10

-8

-6

-4

-2

2

4

6

8

10

12

14

-10 -8 -6 -4 -2 2 4 6 8 10

-14

-12

-10

-8

-6

-4

-2

2

4

6

8

10

Quartic Polynomials

EquationEquation







Factoredy=(x-4)2(x-1)(x+1)

Standardy=x4-8x3+15x2+8x-16

-1,1,4 Positive

As x, y and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -16.95}

Factoredy=-(x-4)2(x-1)(x+1)

Standardy=-x4+8x3-15x2-8x+16

-1,1,4 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 16.95}


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

-15

-12

-9

-6

-3

3

6

9

12

15

18

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

-18

-15

-12

-9

-6

-3

3

6

9

12

15

Quartic Polynomials

EquationEquation







Factoredy=(x+2)3(x-1)

Standardy=x4+5x3+6x2-4x-8

-2,1 Positive

As x, y and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ -8.54}

Factoredy=-(x+2)3(x-1)

Standardy=-x4-5x3-6x2+4x+8

-2,1 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 8.54}


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

-10

-8

-6

-4

-2

2

4

6

8

10

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

-10

-8

-6

-4

-2

2

4

6

8

10

Quartic Polynomials

EquationEquation







Factoredy=(x-3)4

Standardy=x4-12x3+54x2-108x+81

3 Positive

As x, y and

x-, y

Domain

{x| x Є R}

Range

{y| y Є R,

y ≥ 0}

Factoredy=-(x-3)4

Standardy=-x4+12x3-54x2+108x-81

3 Negative

As x, y- and

x-, y-

Domain

{x| x Є R}

Range

{y| y Є R,

y ≤ 0}


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

-10

-8

-6

-4

-2

2

4

6

8

10

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

-10

-8

-6

-4

-2

2

4

6

8

10


• Did we accomplish our objectives?

• Any Questions?



polynomial functions day 1 and 2. polynomial functions do now: find the range of yesterday’s exit...

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