Polynomial Functions

MCT4C1

Polynomial Functions

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

Symmetry in Polynomial Functions

Line symmetry must reflect across y-axis.

Rotational symmetry must rotate 1800 about origin.

Polynomial Function

Degree Type of Symmetry

Type Of Function

y=x2 2 Line Even

y=x3 3 Rotational Odd

y=(x+2)2 2 neither neither

y=x3+1 3 neither neither

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

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 quartic function f(x) = -2x4 + x3 – 5x2 – 10 has a leading

coefficient of -2.

Cubic 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 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

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+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}

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

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-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}

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

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

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

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}

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

-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

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

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}

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

-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

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

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}

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

-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

Factored form & Standard Factored form & Standard formform

X-X-InterceptsIntercepts

Sign of Sign of Leading Leading

CoefficientCoefficient

End End BehaviourBehaviour

Domain and RangeDomain and Range

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}

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

-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

Common Differences

x y

0 1

1 1

2 19

3 79

4 205

5 421

6 751

0

18

60

126

216

330

18

42

66

90

114

24

24

24

24

Since it is a 3rd CommonDifference, the function is CUBIC.

The leading coefficient isPositive.

The leading coefficient canbe found using: 24 = a(3!) 24 = 6a 4 = a