Chapter 4, Problem 33RE

To determine

To analyze: The graph of Rational function $R(x)=\frac{2x-6}{x}$ .

Answer to Problem 33RE

The domain is $\left\{x|x\ne 0\right\}$ .

The vertical asymptote is $x=0$ .

The horizontal asymptote is $y=2$.

Explanation of Solution

Given information:

The rational function $R(x)=\frac{2x-6}{x}$ .

Formula used:

Vertical Asymptote: A rational function $R(x)=\frac{p(x)}{q(x)}$ , in lowest terms, will have a

asymptote $x=r$ if $r$ is a real zero of the denominator q .

Horizontal Asymptote: If a rational function is proper , the line $y=0$ is a asymptote of the graph.

Oblique Asymptote : it occur when the degree of the denominator of a rational

function is one less than the degree of the numerator. The asymptote will be along line

  $y=mx+b$ where $m\ne 0$

Calculation:

Consider ,

  $R(x)=\frac{2x-6}{x}$

Numerator:-

  $\begin{array}{l}=2x-6\\ =2(x-3)\end{array}$

Therefore,

  $R(x)=\frac{2(x-3)}{x}$

Step 1.

The domain of R is the set of all real numbers except $x=0$ .

Step 2.

R is in the lowest terms.

Step 3.

Now, the intercepts of the graph R are as follows:

x -intercept of the graph R

  $\begin{array}{l}=2x-6=0\\ =2x=6\\ =x=\frac{6}{2}\\ =x=3\end{array}$

y -intercept:- There is no , y -intercept as x cannot equal to 0.

Step 4.

Denominator:-

  $=x=0$

The real zeros of the denominator R is $x=0$ . So, the graph of the R has one vertical asymptote $x=0$ .

Step 5.

Since the degree of the numerator ,1,is equal to the degree of denominator 1.

The quotient of R is the leading coefficient of the numerator,2, and the leading coefficient of the denominator ,1.

The horizontal asymptote of the graph R is $y=2$ .

Now, solving the equation we get,

  $R(x)=\frac{2x-6}{x}$

  $\begin{array}{l}=\frac{2x-6}{x}=2\\ =2x-6=2x\\ =2x-2x-6=0\\ =-6\ne 0\end{array}$

This implies the graph does not intersect the line $y=2$ .

Step 6.

  

Figure 1.

Step 7.

Using the information gathered from Step 1 − Step 5 the graph of R has been drawn. The green color shows the graph R(x). The graph has the vertical asymptote at $x=0$ and horizontal asymptote at $y=2$ . The graph does not intersects the line $y=2.$

But intersect x - intercept at point $(3,0)$ .