Question

In Problems $7-44$, follow Steps 1 through 7 on page 228 to analyze the graph of each function.

$R\left(x\right)=\frac{6x^2-7x-3}{2x^2-7x+6}$

Short answer

The graph is

Step-by-step solution

Given information

Given function$R\left(x\right)=\frac{6x^2-7x-3}{2x^2-7x+6}$

Factoring the numerator and denominator

Factoring, we get

$$\begin{gathered} R\left(x\right)=\frac{6x^2-7x-3}{2x^2-7x+6} \\ R\left(x\right)=\frac{6x^2-9x+2x-3}{2x^2-3x-4x+6} \\ R\left(x\right)=\frac{3x(2x-3)+1(2x-3)}{x(2x-3)-2(2x-3)} \\ R\left(x\right)=\frac{(3x+1)(2x-3)}{(x-2)(2x-3)} \\ R\left(x\right)=\frac{3x-1}{x-2} \end{gathered}$$

Calculate the domain

Setting denominator to zero, we get

$$\begin{gathered} (x-2)=0 \\ x=2 \end{gathered}$$

The domain is all real numbers except$2$

Calculate the x  intercept

Setting numerator to zero, we get

$$\begin{gathered} (3x+1)=0 \\ x=-\frac{1}{3} \end{gathered}$$

The$y$intercept is$R\left(0\right)=-\frac{1}{2}$

Calculate the vertical asymptote

The zeroes of denominator is $x=2$so the vertical asymptote is$x=2$

Calculating the horizontal asymptote

The degree of numerator is equal to denominator so the horizontal asymptote is

$$\begin{gathered} y=\frac{a}{b} \\ y=\frac{6}{2} \\ y=3 \end{gathered}$$

$$\begin{gathered} R\left(x\right)=\frac{6x^2-7x-3}{2x^2-7x+6} \\ =3 \\ 6x^2-7x-3=3\left(2x^2-7x+6\right) \\ 6x^2-7x-3=6x^2-21x+18 \\ 14x=21 \\ x=\frac{2}{3} \end{gathered}$$

The intersection point is$(\frac{2}{3},3)$

Sketching the graph

The graph is