The end behavior of a polynomial function is the behavior of the graph the f(x) as x approaches optimistic infinity or negative infinity.

The degree and the leading coefficient of a polynomial duty determine the end behavior of the graph.

The leading coefficient is far-ranging compared to the various other coefficients in the duty for the very huge or very little numbers. So, the sign of the leading coefficient is enough to suspect the end actions of the function.

Degree

Leading Coefficient

End behavior of the function Graph the the role
also optimistic f(x)→+∞,asx→−∞f(x)→+∞,asx→+∞

Example: f(x)=x2

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even negative f(x)→−∞,asx→−∞f(x)→−∞,asx→+∞

Example: f(x)=−x2

*

strange confident f(x)→−∞,asx→−∞f(x)→+∞,asx→+∞

Example: f(x)=x3

*

weird an adverse f(x)→+∞,asx→−∞f(x)→−∞,asx→+∞

Example: f(x)=−x3

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To guess the end-behavior the a polynomial function, first check even if it is the function is odd-degree or even-degree duty and even if it is the leading coefficient is confident or negative.

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Example:

Find the end behavior of the duty x4−4x3+3x+25.

The degree of the duty is even and the top coefficient is positive. So, the end behavior is:

f(x)→+∞,asx→−∞f(x)→+∞,asx→+∞

The graph looks as follows:

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