PreCalculus Class Notes P3 Polynomials: End Behavior, Number/Types of Roots,
Concavity
End Behavior of Polynomial Functions
Let f be a polynomial function with leading coefficient a and degree n.
f(x) = anxn + … + a2x2 + a1x + a0,
anxn is the dominant (leading) term and determines the end behavior of the polynomial
an > 0
an < 0
n≥1
odd
Ends go in
opposite
directions
n≥2
even
Ends go in
same
direction
Positive side
goes up
Positive side
goes down
Types of Roots
y
y
2
y = x−2
Crossing
Root
x
y
2
x
2
y
2
x
3
y
2
x
4
2
x
5
y = ( x − 2)
y = ( x − 2)
y = ( x − 2)
y = ( x − 2)
Tangent
Crossing
Tangent
Crossing
Double root
Triple root
Multiplicity of 4 Multiplicity of 5
Multiplicity of 2 Multiplicity of 3
Concavity
Describes how the curve is changing
Holds
water
Concave up
Dumps
water
Concave down
Point of inflection
Change in concavity
y
y
2
y
2
x
2
x
2
y = x−2
y = ( x − 2)
Constant
Concave up
y
x
3
y = ( x − 2)
Concave down
then concave up
POI
y
2
2
x
y = ( x − 2)
4
x
5
y = ( x − 2)
Concave down
then concave up
POI
Concave up
Characteristics of Polynomials: Standard Form versus Factored Form
Standard Form
Equation
Location of
Roots
Type of roots
y-intercept
Leading term
End behavior
4
3
2
y = x − 3 x − 17 x + 39 x − 20
Factored Form
y = ( x − 5 )( x + 4 )( x − 1)
2
Example: y = ( 2 x − 1)( 3 x + 2 )( 2 − x ) x 2
Location and
multiplicity of
roots
Graphical
description of root
(crossing or
tangent)
y-intercept
Leading term
End behavior
Sketch graph