ZEROS AND END BEHAVIOR
ZEROS AND
END BEHAVIOR
The “zero” of a function is just the value at which a function touches the x-axis.
It is easy to find the roots of a polynomial when it is in factored form!
2 2 15 ( 3)( 5)x x x x
(x - 3) and (x + 5) are factors of the polynomial.
Factored Polynomial
(x - 3) and (x + 5) are factors of the polynomial.
(x - 3)(x + 5) = 0 (we want to know where the polynomial crosses the x-axis)
So (x – 3) = 0 and (x + 5) = 0
The zeros are x = 3, x = -5
Practice: Find the roots of the following factored polynomials.
1. y = (x-2)3(x+3)(x-4)
2. y = (x-5)(x+2)3(x-14)2
3. y = (x+3)(x-15)4
4. y = x2(x+6)(x-6)
Sometimes the polynomial won’t be factored!
Ex. xxxy 623
2nd → TRACE (CALC) → 2: zero
Choose a point to the left of the zero.Then press ENTER.
This arrow indicates that you’ve chosen a
point to the left of the zero.
Choose a point to the rightof the zero.Then press ENTER.
This arrow indicates that you’ve chosen a point to the right of
the zero.
Press ENTER one more time!
Find the zeros of the following polynomials:
13722456317
4562810
234
234
xxxxy
xxxxy
Solutions
4,713722456317
5,1,34562810
234
234
yxxxxy
yxxxxy
End Behavior
The end behavior of a graph describes the far left and the far right portions of the graph.
We can determine the end behaviors of a polynomial using the leading coefficient and the degree of a polynomial.
First determine whether the degree of the polynomial is even or odd.
Next determine whether the leading coefficient is positive or negative.
532)( 2 xxxf degree = 2 so it is even
Leading coefficient = 2 so it is positive
DegreeEven Odd
Lead
ing
Coeffi
cien
t
+
−
High→High Low→High
Low→Low High→Low
Find the end behavior of the following polynomials.
952)( a. 3 xxxf
3624)( b. 24 xxxxf
xxxxf 234)( c. 25
4323)( d. 234 xxxxxf