Name: ______________________
Date: _______________________
Pre Calculus L2
2.2 Notes Day 1
Objectives: By the end of class today you will be able to:
1.) Determine the degree and leading coefficient of a polynomial function.
2.) Determine if a graph is that of a polynomial function.
3.) Use the leading coefficient test to determine the end behavior of a polynomial
function.
Degree: The degree of a polynomial is determined by the greatest exponent of the variable
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The degree of the polynomial f (x) = x5 + 3x2 – 2 is 5
The degree of the polynomial f (x) = x3 – 2x2 + 5x6 is 6
Leading coefficient: The leading coefficient is the number before the variable of the degree of
the polynomial.
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The leading coefficient of the polynomial f (x) = x5 + 3x2 – 2 is 1
The leading coefficient of the polynomial f (x) = x3 – 2x2 + 5x6 is 5
The leading coefficient of the polynomial f (x) = –3x4 +7x2 – 13 is –3
Continuity vs. Discontinuity: The graph of a polynomial function is continuous. This means
that the graph of a polynomial function has no breaks, holes, or gaps.
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A function is continuous if its graph can be drawn with a pencil without
lifting the pencil from the paper.
A function that is discontinuous cannot be drawn without lifting up the
pencil.
This is an example of a polynomial
function; it is continuous.
This is not a polynomial function
because it is discontinuous.
Smooth vs. Jagged: Another feature of a polynomial function is that its graph can only have
smooth, rounded turns. It cannot have sharp turns.
This is an example of a polynomial
function because it has smooth turns.
This is not a polynomial function
because it has a jagged (sharp) turn.
Odd vs. Even degree: Remember the degree of a function is determined by the greatest
exponent.
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If the degree of the function is even, then it is said to have an even degree.
(This is NOT the same as saying that the function is even).
f (x) = x3 – 2x2 + 5x6 has a degree of 6 so the polynomial function has an
even degree.
If the degree of the function is odd, then it is said to have an odd degree.
f (x) = x5 + 3x2 – 2 has a degree of 5 so the polynomial function has an odd
degree.
Leading Coefficient Test: The leading coefficient test determines the end behavior of a
polynomial function.
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If the degree of the function is odd and the leading coefficient is positive
the graph rises to the right and falls to the left.
If the degree of the function is odd and the leading coefficient is negative
the graph rises to the left and falls to the right.
If the degree of the function is even and the leading coefficient is positive
the graph rises right and rises to the left.
If the degree of the function is even and the leading coefficient is negative
the graph falls to the right and falls to the left.
 f (x) = x3
Rises to the right
Falls to the left
 f (x) = – x3
Falls to the right
Rises to the left
 f (x) = x2
Rises to the right
Rises to the left
 f (x) = – x2
Falls to the right
Falls to the left
Example 1: Determine the degree, leading coefficient and end behaviors of the following
polynomial functions.
a. f (x) = x5 + 2x2 – 2 Degree: ______________ Leading Coefficient : _____________
______________ to the right. ______________ to the left.
b. f (x) = 2x4 – 3x3 + 5 Degree: ______________ Leading Coefficient : _____________
______________ to the right. ______________ to the left.
c. f (x) = – 3x7 + 2x3 – x2 + 2 Degree: ___________ Leading Coefficient : _____________
______________ to the right. ______________ to the left.
d. f (x) = – 5x6 + 5x3 – 3 Degree: ______________ Leading Coefficient : _____________
______________ to the right. ______________ to the left.
Classwork: Page 112 #1-8, 9-10 and 15-18
Homework: Finish page 112 classwork.