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 Rule

For a rational function \( f(x)=\frac{N(x)}{D(x)} \), where \( \text{deg}(N(x)) = n \) and \( \text{deg}(D(x)) = m \):

• If \( n < m \), horizontal asymptote is \( y = 0 \).
• If \( n = m \), horizontal asymptote is \( y=\frac{\text{leading coefficient of }N(x)}{\text{leading coefficient of }D(x)} \).
• If \( n > m \), no horizontal asymptote.

For \( f(x)=\frac{4}{x^{2}-3x - 7} \), \( \text{deg}(N(x)) = 0 \) (since \( 4 = 4x^{0} \)) and \( \text{deg}(D(x)) = 2 \). Here, \( 0 < 2 \), so horizontal asymptote is \( y = 0 \).

Step2: Check Intersection with Horizontal Asymptote

To find where the graph crosses \( y = 0 \), set \( f(x)=0 \):

$$ \frac{4}{x^{2}-3x - 7}=0 $$

A fraction equals 0 only when the numerator is 0 (and denominator non - zero). But the numerator is 4, which is never 0. So the graph does not cross \( y = 0 \).

Answer:

a. \( y = 0 \)
b. Graph does not cross \( y = 0 \)

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