Understanding End Behavior of Polynomial Functions
TLDR; Understanding the end behavior of polynomial functions based on degree and leading coefficient.
📈 End Behavior Definition
End behavior or long run behavior of a polynomial function can be determined just by knowing the degree of the polynomial and the sign of the leading coefficient.
It describes the value of the function (or the y value) as x approaches negative or positive infinity.
For even degree polynomials with a positive leading coefficient, as x approaches infinity, f(x) approaches positive infinity.
Similarly, as x approaches negative infinity, f(x) also approaches positive infinity.
If the degree is even and the leading coefficient is negative, the behavior is reversed - as x approaches infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) also approaches negative infinity.
âž• Odd Degree, Positive Coefficient
For odd degree polynomials with a positive leading coefficient, as x approaches infinity, f(x) approaches positive infinity, and as x approaches negative infinity, f(x) approaches negative infinity.
This behavior holds true for polynomials with odd degree and a positive leading coefficient.
âž– Odd Degree, Negative Coefficient
When the degree is odd and the leading coefficient is negative, the behavior is reversed - as x approaches infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches positive infinity.
📉 Graph Analysis
If unsure about the end behavior, it's best to graph the function and analyze the graph for a clearer understanding.