QuestionJune 17, 2025

Find the equation for any horizontal asymptotes for the function below. f(x)=(1-2x+x^2)/(9+2x+3x^2)

Find the equation for any horizontal asymptotes for the function below. f(x)=(1-2x+x^2)/(9+2x+3x^2)
Find the equation for any horizontal asymptotes for the function below.
f(x)=(1-2x+x^2)/(9+2x+3x^2)

Solution
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Answer

y = \frac{1}{3} Explanation 1. Identify the degrees of the polynomials The degree of the numerator 1-2x+x^2 is 2, and the degree of the denominator 9+2x+3x^2 is also 2. 2. Determine horizontal asymptote rule When the degrees are equal, the horizontal asymptote is \frac{a}{b} where a and b are the leading coefficients. 3. Calculate the horizontal asymptote The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 3. Thus, the horizontal asymptote is y = \frac{1}{3}.

Explanation

1. Identify the degrees of the polynomials<br /> The degree of the numerator $1-2x+x^2$ is 2, and the degree of the denominator $9+2x+3x^2$ is also 2.<br />2. Determine horizontal asymptote rule<br /> When the degrees are equal, the horizontal asymptote is $\frac{a}{b}$ where $a$ and $b$ are the leading coefficients.<br />3. Calculate the horizontal asymptote<br /> The leading coefficient of the numerator is 1, and the leading coefficient of the denominator is 3. Thus, the horizontal asymptote is $y = \frac{1}{3}$.
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