End Behavior of Polynomial Functions
f ( x)  3 x 2
End Behavior: Graph rises to the left and graph also rises to the right.
f ( x)  5 x 4
End Behavior: Graph rises to the left and graph also rises to the right.
f ( x)  13x6
End Behavior: Graph rises to the left and graph also rises to the right.
In general, if
f ( x)  a1 x n  a2 x n1  a3 x n2  a4 x n3 
c
has positive leading coefficient (a1 > 0) and the degree of the polynomial is even, then
graph rises to the left and graph also rises to the right.
f ( x)  4 x 2
End Behavior: graph falls to the left and graph also falls to the right.
f ( x)  8 x 4
End Behavior: graph falls to the left and graph also falls to the right.
f ( x)  13x6
End Behavior: graph falls to the left and graph also falls to the right.
In general, if
f ( x)  a1 x n  a2 x n1  a3 x n2  a4 x n3 
c
has negative leading coefficient (a1 < 0) and the degree of the polynomial is even, then
graph falls to the left and graph also falls to the right.
f ( x)  8 x 5  9 x 4  4 x 3  2 x 2  3 x  5
End Behavior: graph falls to the left and graph rises to the right.
In general, if
f ( x)  a1 x n  a2 x n1  a3 x n2  a4 x n3 
c
has positive leading coefficient (a1 > 0) and the degree of the polynomial
is odd, then graph falls to the left and graph rises to the right.
f ( x)  8x5  9 x 4  4 x3  2 x 2  3x  5
End Behavior: graph rises to the left and graph falls to the right.
In general, if
f ( x)  a1 x n  a2 x n1  a3 x n2  a4 x n3 
c
has negative leading coefficient (a1 < 0) and the degree of the polynomial is odd, then
graph rises to the left and graph falls to the right.
Summary of End Behavior of Polynomials
Leading Coefficient
degree of
polynomial
End Behavior
positive
Even
Graph rises to the left and graph rises to the right
negative
Even
Graph falls to the left and graph falls to the right
positive
Odd
Graph falls to the left and graph rises to the right
negative
Odd
Graph rises to the left and graph falls to the right