1.4 Characteristics of Polynomial Functions
Recall:
We can gain insight into the power function's end behaviour by examining the function's leading coefficient and degree.
(erase to reveal.)
(erase to reveal.)
This concept can be extended to include all polynomial functions.
Ex. 1) The degree of the function pictured here is: a) ODD
b) EVEN
Ex. 2) The leading coefficient of the function pictured here is:
a) POSITIVE b) NEGATIVE
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Ex. 3) Choose the two options that describe the end behaviours of the function pictured here:
Ex. 4) For the function, will the graph will have the opposite or the same end behaviours? a) OPPOSITE
b)SAME
Ex. 5) Based on your knowledge of end behaviours, select the graph that represents the function
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Summary of End Behaviors:
1) ODD­degree polynomial functions:
­ opposite end behavior ­ negative leading coefficient:
function extends from quadrant 2 to quadrant 4 positive leading coefficient:
function extends from quadrant 3 to quadrant 1
2) EVEN­degree polynomial functions: ­ the same end behavior
­ negative leading coefficient:
function extends from quadrant 3 to quadrant 4 positive leading coefficient:
function extends from quadrant 2 to quadrant 1
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Terminology
Decreasing and Increasing Functions
­ A function (y­values) is increasing on an interval (x­values) if the graph rises from left to right.
­ A function (y­values) is decreasing on an interval (x­values) if the graph falls from left to right
A turning point is a point at which the function changes
from increasing to decreasing or vice­versa. "Local maximum point" A point (x, y) where the function y = f(x) changes from increasing to decreasing "Local minimum point"
A point (x, y) where the function y = f(x) changes from decreasing to increasing "Absolute maximum point"
The highest point on the graph "Absolute minimum point" The lowest point on the graph.
Ex. 6) Label each turning point:
absolute
absolute
max.
max.
local min.
local max.
absolute
min.
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Key Features
The degree of a polynomial function provides information about the turning points and zeros of a graph.
1) Turning Points:
­ A polynomial function of degree 'n' has at most 'n ­ 1' turning points.
2) Zeros:
­ A polynomial of degree 'n' may have up to 'n' distinct zeros.
­ Odd degree polynomials: must have at least 1 zero
Even degree polynomials: may have no zeros
Other Features
3) Domain and Range:
­ Odd degree polynomials: ­ Even degree polynomials:
4) Maximum and Minimum Points:
­ Odd degree polynomials:
no max. or min. points
­ Even degree polynomials:
At least 1 min. point (positive leading coef.)
At least 1 max. point (negative leading coef.)
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5) Asymptotes:
Graphs of polynomial functions do NOT have vertical or horizontal asymptotes. 6) Symmetry:
­ Odd degree polynomials have 'line symmetry'
­ Even degree polynomials have 'point symmetry'
*Note:
Most polynomial functions have NO symmetrical properties. These functions are neither even nor odd. Ex. 7) For the function f(x) = ­3x5 + 4x3 ­ 8x2 + 7x ­ 5, describe the:
a) End behavior
b) Possible number of turning points
c) Possible number of zeros
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Ex. 8) What could the graph of a polynomial function with a range of and 3 turning points look like?
ppppp
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Table of Values:
General Rule for Finite Differences:
For a polynomial function of degree n, where n is a positive integer, the nth differences
• are equal (constant)
• have the same sign as the leading coefficient
• are equal to a[n x (n ­ 1) x ... x 2 x 1], where a is the leading coefficient
Ex. 9) Complete the following table of values. Use finite differences to determine the:
a) Degree of the polynomial function
b) The sign and value of the leading coefficient
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