Question

what is the horizontal asymptote of the rational function \\( f(x) = \frac{ax^2 + bx + c}{dx^2 + ex + f} \\)? explain. because the degree of the numerator is/are the same as the constant term of the denominator, the horizontal asymptote \\( y = \frac{a}{d} \\) can be found from the ratio of the constant terms. (type an equation.)

Explanation:

Step1: Recall Horizontal Asymptote Rules

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( N(x) \) is the numerator polynomial and \( D(x) \) is the denominator polynomial:

• If the degree of \( N(x) \) (\( \text{deg}(N) \)) is equal to the degree of \( D(x) \) (\( \text{deg}(D) \)), the horizontal asymptote is the ratio of the leading coefficients of \( N(x) \) and \( D(x) \).

Step2: Analyze Given Function

In the function \( f(x)=\frac{ax^{2}+bx + c}{dx^{2}+ex + f} \), the degree of the numerator (\( ax^{2}+bx + c \)) is 2 (since the highest power of \( x \) is 2), and the degree of the denominator (\( dx^{2}+ex + f \)) is also 2. So, \( \text{deg}(N)=\text{deg}(D) = 2 \) (same degree).

Step3: Apply the Rule

When degrees are equal, the horizontal asymptote \( y \) is the ratio of the leading coefficients. The leading coefficient of the numerator is \( a \) (coefficient of \( x^{2} \)) and the leading coefficient of the denominator is \( d \) (coefficient of \( x^{2} \)). Thus, the horizontal asymptote is \( y=\frac{a}{d} \).

Answer:

\( y = \frac{a}{d} \)