Polynomial Functions
PPT 2.3.1
Polynomial Functions
Polynomial
Function in
General Form
y  ax  b
y  ax 2  bx  c
y  ax 3  bx 2  cx  d
y  ax 4  bx 3  cx 2  dx  e
Degree
Name of
Function
1
2
3
4
Linear
Quadratic
Cubic
Quartic
The largest exponent within the polynomial
determines the degree of the polynomial.
Explore Polynomials
Linear
Function
10
8
6
10
8
6
4
4
2
2
-10 -8 -6 -4 -2
2
4
6
8 10
-10 -8 -6 -4 -2
2
-2
-2
-4
-4
Quadratic
Function
-6
-8
-10
10
6
4
2
-10 -8 -6 -4 -2
2
4
6
8 10
-2
-4
-6
-8
-10
6
8 10
-6
-8
-10
10
5
Cubic
Function
8
4
Quartic
Function
-5 -4 -3 -2 -1
-5
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-60
1 2 3 4 5 6 7 8 9 10
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 Polynomials
Look at the two graphs and discuss the questions given below.
Graph A
-10 -8
-6
-4
Graph B
10
10
8
8
6
6
4
4
2
2
-2
2
4
6
8
10
-10 -8
-6
-4
-2
2
-2
-2
-4
-4
-6
-6
-8
-8
-10
-10
4
6
8
10
1. How can you check to see if both graphs are functions?
2. How many x-intercepts do graphs A & B have?
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?
Cubic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form &
Standard form
X-Intercepts
Sign of
Leading
Coefficient
End
Behaviour
12
10
8
Factored
6
4
y=(x+1)(x+4)(x-2)
2
-5
-4
-3
-2
-1
-2
1
2
3
4
5
-4
Standard
-4, -1, 2
Positive
As x,
y and
x-,
y-
Negative
As x,
y-
and
x-,
y
y=x3+3x2-6x-8
-6
-8
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R}
-10
-12
12
10
Factored
8
6
4
y=-(x+1)(x+4)(x-2)
2
-5
-4
-3
-2
-1
-2
-4
-6
-8
-10
-12
1
2
3
4
5
Standard
y=-x3-3x2+6x+8
-4, -1, 2
Domain
{x| x Є R}
Range
{y| y Є R}
Cubic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form &
Standard form
X-Intercepts
Sign of
Leading
Coefficient
End
Behaviour
Domain and Range
12
10
8
Factored
6
4
y=(x+3)2(x-1)
2
-5
-4
-3
-2
-1
-2
1
2
3
4
5
Standard
-4
-6
-3, 1
Positive
y=x3+5x2+3x-9
-8
As x,
y and
x-,
y-
Domain
{x| x Є R}
Range
{y| y Є R}
-10
-12
12
10
Factored
8
6
4
y=-(x+3)2(x-1)
2
-5
-4
-3
-2
-1
-2
1
2
3
4
5
Standard
-4
-6
-8
-10
-12
y=-x3-5x2-3x+9
-3, 1
Negative
As x,
y-
and
x-,
y
Domain
{x| x Є R}
Range
{y| y Є R}
Cubic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form &
Standard form
X-Intercepts
Sign of
Leading
Coefficient
End
Behaviour
12
10
8
Factored
6
4
y=(x-2)3
2
-5 -4 -3 -2 -1
-2
1
2
3
4
5
-4
Standard
2
Positive
y=x3-6x2+12x-8
-6
-8
As x,
y and
x-,
y-
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R}
-10
-12
12
10
Factored
8
6
y=-(x-2)3
4
2
-5 -4 -3 -2 -1
-2
-4
-6
-8
-10
-12
1
2
3
4
5
Standard
y=-x3+6x2-12x+8
2
Negative
As x,
y-
and
x-,
y
Domain
{x| x Є R}
Range
{y| y Є R}
Quartic Polynomials
Look at the two graphs and discuss the questions given below.
Graph A
Graph B
10
14
8
12
6
10
4
8
2
-5
-4
-3
-2
-1
-2
6
1
2
3
4
5
4
2
-4
-6
-5
-4
-3
-2
-8
-10
-12
-14
-1
1
2
3
4
5
-2
-4
-6
-8
-10
1. How can you check to see if both graphs are functions?
2. How many x-intercepts do graphs A & B have?
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?
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard
form
XIntercepts
Sign of
Leading
Coefficient
End
Behaviour
Domain and Range
10
8
6
4
2
-10 -8
-6
-4
-2
-2
2
4
6
8
10
-4
Factored
y=(x-3)(x-1)(x+1)(x+2)
-6
-8
Standard
-10
-12
-2,-1,1,3
Positive
y=x4-x3-7x2+x+6
-14
As x,
y and
x-,
y
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ -12.95}
14
12
10
8
6
4
2
-10 -8
-6
-4
-2
2
-2
-4
4
6
8
10
Factored
y=-(x-3)(x-1)(x+1)(x+2)
-6
-8
-10
Standard
y=-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}
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard
form
XIntercepts
Sign of
Leading
Coefficient
End
Behaviour
15
12
9
Factored
6
3
-5 -4 -3 -2 -1
-3
1
2
3
4
5
-6
y=(x-4)2(x-1)(x+1)
Standard
-9
-12
-1,1,4
Positive
y=x4-8x3+15x2+8x-16
-15
As x,
y and
x-,
y
-18
18
15
12
Factored
9
6
y=-(x-4)2(x-1)(x+1)
3
-5 -4 -3 -2 -1
-3
-6
-9
-12
-15
1
2
3
4
5
Standard
y=-x4+8x3-15x2-8x+16
-1,1,4
Negative
As x,
y-
and
x-,
y-
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ -16.95}
Domain
{x| x Є R}
Range
{y| y Є R,
y ≤ 16.95}
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard
form
XIntercepts
Sign of
Leading
Coefficient
End
Behaviour
10
8
6
Factored
4
2
-5
-4
-3
-2
-1
y=(x+2)3(x-1)
1
2
3
4
5
-2
Standard
-4
-2,1
Positive
y=x4+5x3+6x2-4x-8
-6
-8
As x,
y and
x-,
y
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ -8.54}
-10
10
8
6
Factored
4
y=-(x+2)3(x-1)
2
-5
-4
-3
-2
-1
1
-2
-4
-6
-8
-10
2
3
4
5
Standard
y=-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}
Quartic Polynomials
The following chart shows the properties of the graphs on the left.
Equation
Factored form & Standard
form
XIntercepts
Sign of
Leading
Coefficient
End
Behaviour
10
8
6
Factored
4
y=(x-3)4
2
-5
-4
-3
-2
-1
1
2
3
4
5
-2
-4
Standard
3
Positive
y=x4-12x3+54x2-108x+81
-6
As x,
y and
x-,
y
-8
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ 0}
-10
10
8
6
Factored
4
2
-5
-4
-3
-2
-1
y=-(x-3)4
1
-2
-4
-6
-8
-10
2
3
4
5
Standard
y=-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}