CHARACTERISTICS of POLYNOMIAL FUNCTIONS
(An Introduction)
Polynomial functions of the same degree have similar characteristics.
The degree of a polynomial provides information about the shape,
turning points, and zeros of the graph of the polynomial.
TERMINOLOGY
turning point:
a point on a curve where the function changes from increasing
to decreasing or vice versa
ex. A and B are turning points on the given curve
A

y

B
leading coefficient:
x
the coefficient of the term with the highest degree in a
polynomial
ex. f(x) = 3x4 + 2x3 – 3x + 1 has a leading coefficient of 3
absolute maximum/minimum:
the greatest/least value attained by a function
for all values in its domain
symmetry: a function is odd when it has rotational symmetry about the origin;
all odd functions have the property f(–x) = –f(x)
a function is even when it is symmetric about the y-axis;
all even functions have the property f(–x) = f(x)
Ex 
Recall the three cases for the solution(s) to a quadratic equation:
NO SOLUTION
ONE SOLUTION
TWO SOLUTIONS
(no zeros)
(one zero)
(two zeros)
y
y
x
y
x
x
A quadratic equation has a degree of 2 and may have up to 2 zeros!!
Ex 
Consider the polynomial function f(x) = 3x4 – 4x3 – 4x2 + 5x + 5:
a)
State the leading coefficient.
____________
b)
State the degree of the polynomial.
____________
c)
State the number of zeros.
____________
d)
State the number of turning points.
____________
e)
State the end behaviours (include the quadrant).
____________
f)
____________
Sketch.
INVESTIGATION
Complete the chart using a graphing calculator to aid in the sketches.
NOTE:
A few points about the graphing calculator:
●
●
●
●
press window to set the scale for the x and y axes
press y= to enter each equation and then press graph to view
use ^ for entering exponents
use (–) for a negative value
CHARACTERISTICS of POLYNOMIAL FUNCTIONS
(A Summary)
Number of Zeros
 A polynomial function of degree n may have up to __________ distinct zeros.
 A polynomial of __________ degree must have at least one zero.
 A polynomial of __________ degree may have no zeros.
Turning Points
 A polynomial function of degree n has at most __________ turning points.
End Behaviours
 An odd degree polynomial function has opposite end behaviours.
For a +ve leading coefficient, the function extends from Q ____ to Q ____.
x  , y  ______ and x  –, y  ______
For a –ve leading coefficient, the function extends from Q ____ to Q ____.
x  , y  ______ and x  –, y  ______
 An even degree polynomial function has the same end behaviours.
For a +ve leading coefficient, the function extends from Q ____ to Q ____.
x  ±, y  ______
For a –ve leading coefficient, the function extends from Q ____ to Q ____.
x  ±, y  ______
Ex 
State the degree, leading coefficient and end behaviours for each of the
following polynomial functions:
a)
f(x) = 2x5 – 4x3 + 10x2 – 13x + 8
b)
f(x) = –4x4 + 3x2 – 15x + 5
Ex 
Sketch a graph of a polynomial function that satisfies each set of conditions:
a)




degree 3
–ve leading coefficient
3 zeros
2 turning points
b)




degree 4
+ve leading coefficient
3 zeros
3 turning points
y
y
x
Homework: p.136–138 #1–8, 10–12
x