Question

follow the seven step strategy to graph the following rational function.
$f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}$

a. the y - intercept is \square.
b. there is no y - intercept.
what is/are the x - intercept(s)? select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a.
the x - intercept(s) is/are $1, -\frac{4}{3}$.
(type an integer or a simplified fraction. use a comma to separate answers as needed.)
b. there is no x - intercept.
find the vertical asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a.
the equation(s) of the vertical asymptote(s) is/are $x = 0, x = \frac{7}{2}$.
(type an equation. use a comma to separate answers as needed.)
b. there is no vertical asymptote.
find the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the equation(s) of the horizontal asymptote(s) is/are \square.
(type an equation. use a comma to separate answers as needed.)
b. there is no horizontal asymptote.

Explanation:

Step1: Find the horizontal asymptote

For a rational function \(f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), if \(n = m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator), the horizontal asymptote is \(y=\frac{a_n}{b_m}\).
Here, the numerator \(3x^{2}+x - 4\) has degree \(n = 2\) and the denominator \(2x^{2}-7x\) has degree \(m=2\).
So, \(a_n = 3\) and \(b_m=2\).
The equation of the horizontal asymptote is \(y=\frac{3}{2}\)

Answer:

A. The equation(s) of the horizontal asymptote(s) is/are \(y = \frac{3}{2}\)