Question

find the horizontal asymptote of the graph of the rational function.
$y = \frac{x + 4}{4x^2 - 7}$
identify the horizontal asymptote for the graph of the function.
\\(\bigcirc\\) $y = \frac{1}{4}$
\\(\bigcirc\\) there is no horizontal asymptote.
\\(\bigcirc\\) $y = 0$
\\(\bigcirc\\) $y = -4$

Explanation:

Step1: Recall horizontal asymptote rules

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

• If the degree of \( f(x) \) is less than the degree of \( g(x) \), the horizontal asymptote is \( y = 0 \).
• If the degrees are equal, the horizontal asymptote is \( y=\frac{\text{leading coefficient of }f(x)}{\text{leading coefficient of }g(x)} \).
• If the degree of \( f(x) \) is greater than the degree of \( g(x) \), there is no horizontal asymptote (but there may be an oblique asymptote).

Step2: Determine degrees of numerator and denominator

For the function \( y=\frac{x + 4}{4x^{2}-7} \):

• The numerator \( f(x)=x + 4 \) is a polynomial of degree \( 1 \) (the highest power of \( x \) is \( 1 \)).
• The denominator \( g(x)=4x^{2}-7 \) is a polynomial of degree \( 2 \) (the highest power of \( x \) is \( 2 \)).

Since the degree of the numerator (\( 1 \)) is less than the degree of the denominator (\( 2 \)), by the rule for horizontal asymptotes, the horizontal asymptote is \( y = 0 \).

Answer:

\( y = 0 \)