Question

a. identify the horizontal asymptote (if any).
b. if the graph of the function has a horizontal asymptote, determine the point where the graph crosses the horizontal asymptote.

$f(x) = \frac{4}{x^2 - 3x - 7}$

\bigcirc a. $y = 0$
b. $(0, 0)$

\bigcirc a. $y = 4$
b. $(0, 4)$

\bigcirc a. $y = 0$
b. graph does not cross $y = 0$.

\bigcirc a. no horizontal asymptote
b. not applicable

Explanation:

Step1: Analyze Horizontal Asymptote

For a rational function \( f(x) = \frac{N(x)}{D(x)} \), when the degree of \( N(x) \) (let's call it \( n \)) is less than the degree of \( D(x) \) (let's call it \( d \)), the horizontal asymptote is \( y = 0 \). Here, \( N(x) = 4 \) (degree \( n = 0 \)) and \( D(x)=x^2 - 3x - 7 \) (degree \( d = 2 \)). Since \( n < d \), horizontal asymptote is \( y = 0 \).

Step2: Check Intersection with Asymptote

To find where the graph crosses \( y = 0 \), set \( f(x)=0 \), i.e., \( \frac{4}{x^2 - 3x - 7}=0 \). But a fraction is zero only when numerator is zero (and denominator non - zero). Here numerator is 4, which is never zero. So the graph does not cross \( y = 0 \).

Answer:

a. \( y = 0 \)
b. Graph does not cross \( y = 0 \) (The corresponding option is: a. \( y = 0 \); b. Graph does not cross \( y = 0 \))