In polynomials of degree 2 or higher, both ends of the function will eventually either rise or fall exponentially. The directions that each arm of the graph extends are known as the graph's end behavior. End behavior can be easily identified from the leading coefficient and the degree of the leading term.
Identify the Leading Term
In a polynomial, a term is a part of a polynomial separated from other terms by + and - signs. The leading term is the term that contains the variable to the highest power. For example, in the polynomial "8x^6 + 2x^2 + x - 7," there are four terms, and the term "8x^6" is the leading term because the exponent, 6, is the greatest variable exponent in the polynomial.
The leading term determines the end behavior of the polynomial. The coefficient of the leading term is called the leading coefficient. For example, if "8x^6" is the leading term, 8 is the leading coefficient.
End Behavior of the Right Side
On the right side of a polynomial graph, as x increases towards infinity, the graph will either increase or decrease at an exponential rate. This is determined by the leading coefficient. If the leading coefficient is positive, the graph will increase exponentially. If the leading coefficient is negative, the graph will decrease exponentially. For example, in the polynomial 2x^3 + x - 9, the right side of the polynomial will increase exponentially because the leading coefficient, 2, is positive.
End Behavior of the Left Side
The end behavior of the left side of a polynomial, as x decreases towards negative infinity, is determined by the degree of the leading term and the exponent, or degree, of the leading term. If the leading term is of even degree, the end behavior of the left side of the polynomial will mirror the right side. If the leading term is of odd degree, the end behavior of the left side of the polynomial will be the opposite of the right side. For example, if "ax^4" is the leading term, the end behavior of the left side will match that of the right because 4 is even. If the leading term was "ax^5," the end behavior of the left side would be opposite that of the right.
Graphing End Behavior
To represent the end behavior of a polynomial on a graph, place arrow heads on the ends of the polynomial after graphing enough that it is clear which direction the end of the polynomial is going. This is similar to graphing the end behavior of a line.
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