Question

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

( type an integer or a simplified fraction. use a comma to separate answers as needed. )
\bigcirc 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.
\bigcirc \boldsymbol{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. )
\bigcirc 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.
\bigcirc \boldsymbol{a}.
the equation(s) of the horizontal asymptote(s) is/are $y = \frac{3}{2}$.
( type an equation. use a comma to separate answers as needed. )
\bigcirc b. there is no horizontal asymptote.

plot points between and beyond each x-intercept and vertical asymptote. find the value of the function at the given value of x.

$x$	$-3$	$-1$	$\frac{1}{2}$	$5$	$6$

( simplify your answers. )

Explanation:

Step1: Substitute \(x = - 3\) into \(f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}\)

$$
LATEXBLOCK0
$$

Step2: Substitute \(x=-1\) into \(f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}\)

$$
LATEXBLOCK1
$$

Step3: Substitute \(x = \frac{1}{2}\) into \(f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}\)

$$
LATEXBLOCK2
$$

Step4: Substitute \(x = 5\) into \(f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}\)

$$
LATEXBLOCK3
$$

Step5: Substitute \(x = 6\) into \(f(x)=\frac{3x^{2}+x - 4}{2x^{2}-7x}\)

$$
LATEXBLOCK4
$$

Answer:

When \(x=-3\), \(f(x)=\frac{20}{39}\); when \(x = - 1\), \(f(x)=-\frac{2}{9}\); when \(x=\frac{1}{2}\), \(f(x)=\frac{11}{12}\); when \(x = 5\), \(f(x)=\frac{76}{15}\); when \(x = 6\), \(f(x)=\frac{11}{3}\)