Algebra 2 Trig
Unit 5
Name:_________________________________ Date:_____________
Definition:
Let n be a nonnegative integer and let an, an-1, . . . , a2, a1, a0 be real numbers with an ≠ 0. The function given by
𝑓(𝑥) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0
is called a ________________________ function of ___ with degree ___.

Polynomial functions are always ___________________.
This means:

Polynomial functions always have __________________________ curves.
This means:
*When n is an even integer (greater than zero), graph is similar to _____________.
*When n is an odd integer, graph is similar to _____________.
_______________________________: 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) -5x + 7x – 2x
degree: ________
leading coefficient: ________
3) 7x – 13x + x – 1
degree: ________
leading coefficient: ________
3
2
5
3
End Behavior
 The behavior of the graph as x approaches positive infinity (+∞) or negative infinity (−∞).
 The expression 𝑥 → +∞ is read as “x approaches positive infinity.”
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).
As the polynomial graph moves from −∞ 𝑡𝑜 ∞, 𝑓(𝑥) eventually rises or falls in the following manner:
n is ODD
Positive (+)
LC
Negative (-)
LC
1) (𝑥) = 𝑥 4 − 5𝑥 2 + 4
n is EVEN
Positive (+)
LC
Negative (-)
LC
2) 𝑓(𝑥) = 4𝑥 − 𝑥 3
3) 𝑓(𝑥) = 3𝑥 5 − 2𝑥 4 + 9𝑥 − 1
4) 𝑓(𝑥) = −2𝑥 6 + 5
DEGREE = # OF COMPLEX ROOTS
*The following examples all deal with POSITIVE leading coefficients.
degree 1
degree 2
degree 3
degree 4
degree 5
functions w/ odd degrees:
functions w/ even degrees:


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.
(test points in between each x-intercept if you still are unsure of how to graph)
EXAMPLES:
5) f(x) = 2x3 – 6x2
zeros: ___________ y-int: ______
6) f(x) = -x4 + 4x2
7) f(x) = x3 – 2x2 – x + 2
zeros: ___________ y-int: ______
8) f(x) = x4 – 10x2 + 9
zeros: ___________ y-int: ______
zeros: ____________ y-int: ______
9) f(x) = -x5 + 5x3 – 4x
zeros: ___________ y-int: ______
Given the following roots, write a polynomial function.
10) roots = 4, -2
11) roots = -1, 2, 1
12) roots = 7i, - 7i