Algebra 2
5.3 Discovery
Name_________________________
OBJECTIVE: To understand what the graphs of polynomial functions will look like.
 Predict the maximum number of turns possible for the polynomial
 Accurately predict the left and right end behavior of the graph of any polynomial.
Vocabulary
1)
2)
3)
The degree of a polynomial is the highest exponent in the polynomial
The leading coefficient is the coefficient of the term with the highest degree
Standard form for a polynomial arranges the terms by degree, from highest to lowest
Type of Function
Equation of a line
Equation of a quadratic
Equation of a polynomial
General Form of Equation
Degree
Leading
Coefficient
y = mx + b
y = ax2 + bx + c
y = -3x3 – 2x2 + 3x – 5
Using your graphing calculator graph each of the following equations. Complete the table and try to find any
patterns regarding the behavior (shape) of the graph.
Equation
y = -x3 + 4x
y = x4 – 5x2 + 4
y = x3 – 9x
y = x4 – 4x2 + x
y = -x4 + 3x2 + 2x +4
y = x5 – 3x3 + 2
Graph
Degree of
polynomial
# of Turns
(look at
graph)
Left end
behavior
(as you go to
the left the
picture is..)
Leading
Coefficient
Right end
behavior
(as you go to
the right the
picture is..)
rising
positive
rising
falling
negative
falling
rising
positive
rising
falling
negative
falling
rising
positive
rising
falling
negative
falling
rising
positive
rising
falling
negative
falling
rising
positive
rising
falling
negative
falling
rising
positive
rising
falling
negative
falling
If the leading coefficient is positive, and the degree is even, then the graphs at the ends will resemble__________
If the leading coefficient is positive, and the degree is odd, then the graphs at the ends will resemble___________
If the leading coefficient is negative, and the degree is even, then the graphs at the ends will resemble__________
If the leading coefficient is negative, and the degree is odd, then the graphs at the ends will resemble__________
If the degree is n, then the graph has at most ____________ turns.
1. Use the rules discovered on the front side of this sheet to answer the following questions.
2. Note that there may be more than one solution to a problem. List all the answers in alphabetical order.
3. If there is no solutions, write “NO SOLUTION”
A)
B)
C)
D)
E)
F)
1.
Which of the graphs above are of polynomials with a positive leading
coefficient?
__________
2.
Which of the graphs above are of polynomials with positive leading
coefficient AND an even degree?
__________
3.
Which of the graphs above are of polynomials with an odd degree?
__________
4.
Which of the graphs above are of polynomials with negative leading
coefficient AND an odd degree?
__________
5.
Which graphs have to be polynomials of at least degree 4?
__________
6.
From the picture, how many roots or zeroes does equation “d” have?
__________
7.
From the picture, how many y-intercepts does equation “a” have?
__________
8.
For the equation y  237 x 77  14 x 3 , the degree is _________ and the left end behavior is _____________ while
the right end behavior is ____________ with the maximum number of turns possible being ___________..