Transcript Polynomial Functions

Polynomial Functions
PPT 2.3.1
Polynomial Functions
Polynomial
Function in
General Form
y  ax  b
y  ax2  bx  c
y  ax3  bx2  cx  d
y  ax4  bx3  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
Quadratic
Function
Cubic
Function
Quartic
Function
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
Graph B
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
Factored
y=(x+1)(x+4)(x-2)
Standard
-4, -1, 2
Positive
As x,
y and
x-,
y-
Negative
As x,
y-
and
x-,
y
y=x3+3x2-6x-8
Factored
y=-(x+1)(x+4)(x-2)
Standard
y=-x3-3x2+6x+8
-4, -1, 2
End
Behaviour
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R}
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
Factored
y=(x+3)2(x-1)
Standard
-3, 1
Positive
y=x3+5x2+3x-9
Factored
y=-(x+3)2(x-1)
Standard
y=-x3-5x2-3x+9
-3, 1
Negative
End
Behaviour
As x,
y and
x-,
y-
As x,
y-
and
x-,
y
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R}
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
Factored
y=(x-2)3
Standard
2
Positive
y=x3-6x2+12x-8
Factored
y=-(x-2)3
Standard
y=-x3+6x2-12x+8
2
Negative
End
Behaviour
As x,
y and
x-,
y-
As x,
y-
and
x-,
y
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R}
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
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
Factored
y=(x-3)(x-1)(x+1)(x+2)
Standard
-2,-1,1,3
Positive
y=x4-x3-7x2+x+6
Factored
y=-(x-3)(x-1)(x+1)(x+2)
Standard
y=-x4+x3+7x2-x-6
-2,-1,1,3
Negative
End
Behaviour
As x,
y and
x-,
y
As x,
y-
and
x-,
y-
Domain and Range
Domain
{x| x Є R}
Range
{y| y Є R,
y ≥ -12.95}
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
Factored
y=(x-4)2(x-1)(x+1)
Standard
-1,1,4
Positive
y=x4-8x3+15x2+8x-16
Factored
y=-(x-4)2(x-1)(x+1)
Standard
y=-x4+8x3-15x2-8x+16
-1,1,4
Negative
End
Behaviour
As x,
y and
x-,
y
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
As x,
y and
x-,
y
-8
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}