Question

How does the degree of a polynomial affect its end behavior?

Answer 1

Hint: The end behavior of a function describes the behavior of a graph of a function at the ends of the \[x\] -axis. The degree of the polynomial is the highest power or exponent of the variable of the polynomial. Using these definitions, we will find how the end behavior of a polynomial is affected by the degree of the polynomial.

Complete Step by Step Solution:
We know that the end behavior of a function describes the graph to the right end of the \[x\]- axis tends to\[ + \infty \]and the graph to the left end of the \[x\]- axis tends to \[ - \infty \].
We know that a polynomial function is defined as the sum of terms which is in the form \[a{x^n}\] where \[a,x,n\] be the real number, variable and an integer respectively.
The end behavior of a Polynomial is determined by its degree and the leading coefficient. We should follow these rules to find the effect of the end behavior by the degree of the Polynomial as:
1. If the degree is even and the leading coefficient is positive, then both the ends of the graph for the function will point up.
2. If the degree is even and the leading coefficient is negative, then both the ends of the graph for the function will point down.
3. If the degree is odd and the leading coefficient is positive, then the right end of the graph for the function will point up and the left end of the graph for the function will point down.
4. If the degree is odd and the leading coefficient is negative, then the right end of the graph for the function will point down and the left end of the graph for the function will point up.
Therefore, these are the end behavior of a polynomial that has been affected by the degree of the polynomial.

Note:
We know that the leading coefficient of the polynomial is defined as the coefficient corresponding to the highest degree in the given polynomial. A polynomial function is defined as a function which has more than three terms. There are different types of polynomials based on the highest degree of variable. End behavior of a function is applicable for all types of function.