End Behavior of a polynomial
End Behavior of a polynomial function can be determined from the characteristics of the Lead Term.
Lead Term will fall into one of the 4 categories:
Degree: Odd
Coefficient: Positive
Down left
Degree: Even
Degree: Odd
Up right
Coefficient: Positive
Up left
Coefficient: Negative
Up left
Degree: Even
Up right
Down right
Coefficient: Negative
Down left
Down right
Multiplicity of a “zero” (of an x-intercept)
The “c” is a zero of a function, then the multiplicity of “c” is the number of times that “c” is a zero for the function.
Ex. y = ( x − 4)( x + 2)3 To find the zeros we let “y” =0 and solve for x.
0 = ( x − 4)( x + 2)3
so, x = 4 or x = −2
In this case we would say that 4 is a zero of multiplicity 1 because is came from the factor ( x − 4 ) (Note that factor is to 1st power).
Similarly, “- 2” is a zero of multiplicity 3 because it came from the factor ( x + 2) (factor is to 3rd power).
3
If the zero is a real number, then it will be an x-intercept.
Multiplicity of a zero is EVEN Î graph will BOUNCE the x-axis at that x-intercept
Multiplicity of a zero is ODD Î graph will CROSS the x-axis at that x-intercept
Ex. Find the zeros and their multiplicities
3
y = x( x + 3) 2 ( x − 1)
0 = x( x + 3) 2 ( x − 1)
2
3
so,
x=0
mult 1
or x = −3 or x = 1
mult 2
1
mult 3
x = 0 is a zero of mult 1 ⇒ graph crosses x-axis at (0,0)
x = −3 is a zero of mult 2 ⇒ graph bounces off x-axis at (-3,0)
-2
2
x = 1 is a zero of mult 3 ⇒ graph crosses x-axis at (1,0)
-1