Hmwk: pg 136#1-3, 4abcd, 5, 6ac, 7, 8, 10ab, 11, 13, (16)
Characteristics of Polynomial Functions
Polynomial functions have common features depending on the sign of the leading
coefficient and the degree.
Leading Coefficient - the coefficient of the term with the highest degree
in a polynomial; usually it is the first coefficient.
1. Examine the following functions and state their degree.
a)
f ( x)  x 3  2 x
b)
f ( x)  2 x5  7 x4  3x3  18x 2  5
degree = __
c) f ( x)  2 x3  4 x2  3x  1
degree = __
degree = __
d)
f ( x)  5 x 5  5 x 4  2 x 3  4 x 2  3 x
degree = __
All of the functions above have an _____ degree.
Key Features of Odd Degree Functions
End behaviours
 If the leading coefficient is positive, then the function extends from the ______
quadrant to the _______ quadrant; ie as x   , y   and as x   , y   .
 If the leading coefficient is negative, then the function extends from the ___
quadrant to ________ quadrant; ie as x   , y   and as x   , y   .
Turning Points
 These polynomials will have at most "n - 1" turning points; notice in 'd)' that
there are only ____ turning points even though the function is of degree ___.
Number of Zeroes (x-intercepts)
 They will have at least ____ x-intercept with a maximum of 'n' x-intercepts.
2. Examine the following functions and state their degree.
a) f ( x)  3x4  4 x3  4 x2  5x  5
b)
f ( x)  2 x6  12 x 4  18x 2  x  10
degree = ___
c) f ( x)   x4  2 x3  x2  2 x
degree = ___
degree = ___
d)
f ( x)   x 2  2 x
degree = ___
All of the functions above have an _____ degree.
Key Features of Even Degree Functions
End behaviours
 If the leading coefficient is positive, then the function extends from the ______
quadrant to the _______ quadrant; ie as x   , y   and as x   , y   .
 If the leading coefficient is negative, then the function extends from the ___
quadrant to ________ quadrant; ie as x   , y   and as x   , y   .
Turning Points
 These polynomials will have at most "n - 1" turning points.
Number of Zeroes (x-intercepts)
 They may not have any x-intercepts but can have a maximum of 'n' x-intercepts.
Symmetry of Polynomial Functions
Polynomial functions can have odd, even or no symmetry. If a polynomial does have
symmetry, it tends to follow the degree. ie; If a quartic function (degree 4) has
symmetry then it will be even since the degree is even, but not all quartics have
symmetry.
***CAREFUL!! While the symmetry and degree of a function are similar, they
are not necessarily the same concept. ***
Practice
Given the followig functions, complete the tables below.
a) f ( x)  2 x5  3x3  x  8
b) f ( x)  x 4  6 x3  2 x
c) f ( x)   x6  x5  x  5
Degree
Degree
Degree
Even or Odd
Degree
Sign of Leading
Coefficient
Max # of
Turning Points
Max # of
x-ints
End Behaviours
Even or Odd
Degree
Sign of Leading
Coefficient
Max # of
Turning Points
Max # of
x-ints
End Behaviours
Even or Odd
Degree
Sign of Leading
Coefficient
Max # of
Turning Points
Max # of
x-ints
End Behaviours