How to Find the End Behavior of Polynomials?

To predict the end behavior of a polynomial function, first, check whether the function is an odd-degree or even-degree function and whether the leading coefficient is positive or negative.

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A step-by-step guide to end behavior of polynomials

The end behavior of a polynomial function describes how the graph behaves as \(x\) approaches \(±∞\). We can determine the end behavior by looking at the leading term (the term with the highest \(n\)-value for \(ax^n\), where \(n\) is a positive integer and \(a\) is any nonzero number) of the function.

The leading coefficient is significant compared to the other coefficients in the function for the very large or very small numbers. Therefore, the leading coefficient sign is sufficient to predict the end behavior of the function.

Depending on the sign of the coefficient \((a)\) and the parity of the exponent \((n)\), the end behavior differs:

End Behavior of Polynomials , Example 1:

Find the end behavior of the function \(f(x)= x^4-4x^3+3x+25\).

Solution:

The degree of the function is even and the leading coefficient is positive. So, the end behavior is:

\(f(x)\)→\(+∞\), as \(x\) →\(−∞\)

\(f(x)\)→\(+∞\), as \(x\) →\(+∞\)

Exercises for End Behavior of Polynomials

Find the end behavior of each function.

1. \(\color{blue}{f(x)=-x^5+4x^3-9x, x→−∞}\)
2. \(\color{blue}{f(x)=6x^6-4x^4+2x-3, x→+∞}\)
3. \(\color{blue}{f(x)=(x+4)^2+x^4+3, x→-∞}\)
4. \(\color{blue}{f(x)=-8x^4+3x^3+11x^2+7, x→+∞}\)

1. \(\color{blue}{f(x)→+∞}\)
2. \(\color{blue}{f(x)→+∞}\)
3. \(\color{blue}{f(x)→+∞}\)
4. \(\color{blue}{f(x)→-∞}\)