Algebra 2/Trig
Date:_____________
Name:_________________________________
A ___________________________________ is an expression of more than two algebraic terms.

Polynomial functions are always ___________________.
This means: _________________________________________________

Polynomial functions always have _____________________________ curves.
This means: _______________________________________________________
*When the exponent is an even integer (greater than zero), graph is similar to _____________.
*When the exponent is an odd integer, graph is similar to _____________.
Ex:
Ex:
_______________________________: coefficient of the term with the greatest exponent
For each of the following polynomials, state the degree and the leading coefficient.
1) x4 + x2 – x + 1
degree: ________
leading coefficient: ________
2) -5x3 + 7x2 – 2x
degree: ________
leading coefficient: ________
End Behavior
 The behavior of the graph as _____ approaches positive infinity (+∞) or negative infinity (−∞).
 Think: the direction of the ENDS of the graph (Starts ____, Ends_____)
* Leading Coefficient Test

The leading coefficient of a polynomial function can tell you the end behavior of each graph
(whether the graph rises or falls from left to right).
 “n” is the EXPONENT
n is ODD; ___________
Positive (+)
LC
Negative (-)
LC
1) (𝑥) = 𝑥 4 − 5𝑥 2 + 4
2) 𝑓(𝑥) = 4𝑥 − 𝑥 3
n is EVEN:
Positive (+)
LC
Negative (-)
LC
3) 𝑓(𝑥) = 3𝑥 5 − 2𝑥 4 + 9𝑥 − 1
4) 𝑓(𝑥) = −2𝑥 6 + 5
Recall: The DEGREE is the number of complex roots (total!)
DOUBLE ROOT: _________________________________________
On a graph, think “____________________________”
IMAGINARY ROOTS: ______________________________________
On a graph, they don’t cross the ____________________________
ODD degrees: must cross the x–axis ________or any ODD number of times ____degree
EVEN degrees: may or may not cross x–axis (0), but if it does, it crosses an even # of times ____degree
*The following examples all deal with POSITIVE leading coefficients.
degree 1
degree 2
degree 3
degree 4
degree 5
**Number of TURNS = ____________________________________________________****
Put it together: Given the polynomial, give the degree, Leading Coefficient, End behavior, and possible # turns
1) f(x) = 4x5 – 12x2
Degree:_________
LC: ________ End Behavior:____________ # turns:_______
2) f(x) = -7x6 + 9x
Degree:_________
LC: ________ End Behavior:____________ # turns:_______
3) f(x) = -x4 +3x - 1
Degree:_________
LC: ________ End Behavior:____________ # turns:_______


Functions of degree greater than 2 are much more complicated to graph.
Therefore, the graphs will not be as specific.
When n is greater than 2, use leading coefficient test, then plot zeros and y-intercept
EXAMPLES:
1) f(x) = 2x3 – 6x2
# turns:_____ End Beh:_______
zeros: ___________ y-int: ______
4) f(x) = x4 – 10x2 + 9
# turns:_____ End Beh:_______
zeros: __________ y-int: ______
2) f(x) = -x4 + 4x2
# turns:_____ End Beh:_______
zeros: ___________ y-int: ______
5) f(x) = -x5 + 5x3 – 4x
# turns:_____ End Beh:_______
zeros: __________ y-int: ______
Given the following roots, write a polynomial function.
7) roots = 4, -2
8) roots = -1, 2, 1
3) f(x) = x3 – 2x2 – x + 2
# turns:_____ End Beh:_______
zeros: __________ y-int: ______
6) f(x) = x5 + 2x4 – 4x3 – 8x2
# turns:_____ End Beh:_______
zeros: ____________ y-int: ______
9) roots = 7i, - 7i