The end actions of a polynomial duty is the actions of the graph that f(x) together x approaches optimistic infinity or negative infinity.

The degree and the top coefficient of a polynomial function determine the end behavior of the graph.

The top coefficient is significant compared to the other coefficients in the role for the very big or very small numbers. So, the sign of the top coefficient is sufficient to suspect the end habits of the function.

Degree
 Leading Coefficient End actions of the role Graph the the duty even hopeful f(x)→+∞,asx→−∞f(x)→+∞,asx→+∞ Example: f(x)=x2 even negative f(x)→−∞,asx→−∞f(x)→−∞,asx→+∞ Example: f(x)=−x2 strange hopeful f(x)→−∞,asx→−∞f(x)→+∞,asx→+∞ Example: f(x)=x3 odd an unfavorable f(x)→+∞,asx→−∞f(x)→−∞,asx→+∞ Example: f(x)=−x3

To guess the end-behavior of a polynomial function, first check whether the role is odd-degree or even-degree function and whether the leading coefficient is hopeful or negative.

You are watching: What can you say about the end behavior of the function ?

Example:

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

The level of the function is even and the leading coefficient is positive. So, the end actions is:

f(x)→+∞,asx→−∞f(x)→+∞,asx→+∞

The graph looks together follows:

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