Question

follow the steps for graphing a rational function to graph the function ( r(x)=\frac{5 x^{2}-7 x - 3}{2 x^{2}-7 x + 6} ).

b. it approaches ( infty ) on one side of the asymptote(s) at ( x = 2 ) and ( -infty ) on the other.
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)

c. it approaches either ( infty ) or ( -infty ) on both sides of the asymptote(s) at ( x=).
(type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.)

d. the function has no vertical asymptote.

determine the horizontal asymptote(s), if any exist. select the correct choice and, if necessary, fill in the answer box(es) to complete your choice.

a. the function has one horizontal asymptote.
(type an equation. use integers or fractions for any numbers in the equation.)

b. the function has two horizontal asymptotes. the top asymptote is and the bottom asymptote is
(type equations. use integers or fractions for any numbers in the equations.)

c. the function has no horizontal asymptote.

Explanation:

Step1: Find the degree of numerator and denominator

The degree of the numerator \(5x^{2}-7x - 3\) is \(n = 2\) (since the highest - power term is \(5x^{2}\)).
The degree of the denominator \(2x^{2}-7x + 6\) is \(m=2\).

Step2: Use the rule for horizontal asymptotes

When \(n = m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator) for a rational function \(R(x)=\frac{a_{n}x^{n}+\cdots+a_{0}}{b_{m}x^{m}+\cdots + b_{0}}\), the horizontal asymptote is given by \(y=\frac{a_{n}}{b_{m}}\).
Here, \(a_{n}=5\) (the coefficient of \(x^{2}\) in the numerator) and \(b_{m}=2\) (the coefficient of \(x^{2}\) in the denominator).

Answer:

A. The function has one horizontal asymptote. \(y = \frac{5}{2}\)