Final answer:
To determine the end behavior of a rational function, compare the degrees of the numerator and the denominator to predict how the function behaves as x approaches infinity. Horizontal asymptotes and the ratio of the leading coefficients further inform the end behavior. Checking the reasonableness ensures consistency with established behavior of rational functions.
Step-by-step explanation:
Understanding the End Behavior of Rational Functions
To determine the end behavior of a rational function, you essentially want to understand how the function behaves as the input (usually x) approaches infinity or negative infinity. The end behavior of a rational function is determined by the degrees and leading coefficients of the numerator and denominator polynomials.
- First, identify the highest power of x in both the numerator and the denominator.
- Compare the degrees of these terms. If the degree of the numerator is greater than the degree of the denominator, the end behavior will resemble that of the leading term of the numerator (since it will dominate as x becomes very large).
- If the degree of the denominator is greater, then the function will approach zero as x approaches both positive and negative infinity.
- If the degrees are the same, the end behavior of the function will approach the ratio of the leading coefficients of the numerator to the denominator.
- Also consider any horizontal asymptotes, which can be found when the degrees are the same; the y-value of the asymptote will be this ratio of leading coefficients.
- Finally, check the reasonableness of your end behavior conclusion. It should be consistent with the general rules of rational functions and their asymptotic behavior.
Through this process, you can predict whether the function will rise to infinity, fall to negative infinity, or level off at a horizontal asymptote as x becomes very large in both the positive and negative directions.