2.2 Polynomial Functions of Higher Degree

Graphs, Intercepts, and Extrema

Graphs

Here are some examples of graphs of polynomials, we can use these to help illustrate the relationship between degree of polynomials and the number of intercepts or extrema.

Zero Degree Polynomial

First Degree Polynomial

Second Degree Polynomial

Third Degree Polynomial

Fourth Degree Polynomial

Fifth Degree Polynomial

Intercepts

To find the number of intercepts in a polynomial function one can use the graph of the function or use the degree of the polynomial. When we look at the graphs the number of intercepts is either equal or less than the degree of the polynomial (when there are zeros of 2 or 3 multiplicity).

Number of intercepts (i) of a polynomial of degree (n) can be found with  i ≤ n

Extrema

To find the number of maxima and minima of a polynomial one can use to the graph of the function or use the relationship between degree and number of extrema.

Number of extrema (e) of a polynomial of degree (n) be found with e = n-1

Limit Notation

Limit Notation is used to describe End Behavior, or how a function behaves as it approaches ∞ or -∞.

Example for f(x)=-5x^2+23 

Left End:                                                          Right End:

lim f(x)=-∞                                                       lim f(x)=∞

x -> ∞                                                               x -> ∞

"The limit of f(x) as x approaches infinity"

In order to predict end behavior one can calculate values for a massive positive number then calculate the value of a very large negative number. Also one can use the characteristics of odd and even graphs to predict end behavior.

When the highest degree is Even: limits will be identical

When the highest degree is Odd: limits will be opposite

Steps:

1. Find term with largest degree

2. Look at the coefficient in front of term

If the coefficient is negative and the degree is even, both limits are -∞

If the coefficient is negative and the degree is odd, the right limit is -∞ and the left limit is ∞

If the coefficient is positive and the degree is even, both limits are ∞

If the coefficient is positive and the degree is odd, the right limit is ∞ and the left limit is -∞

That is all.

Tommy McLeod