Question

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

$R\left(x\right)=\frac{2x+4}{x-1}$

Short answer

The graph for the function $R\left(x\right)=\frac{2x+4}{x-1}$is,

Step-by-step solution

Step 1 Given function is,

$R\left(x\right)=\frac{2x+4}{x-1}$

The domain of $R$is role="math" localid="1646228446854" $\left\{x|x\ne 1\right\}$.

Also, $R$is in the lowest terms.

The $x$- intercept can be found by determining the real zeroes of the numerator of $R$which is written in the lowest terms.

Therefor, by solving $x=-2$.

The $x$- intercept is, $-2$.

The $y$- intercept can be found by determining the value of $R$when $x=0$.

The value of$R$is$-4$when$x=0$. So, the$y$- intercept is,$-4$.

Step 2 We can find the vertical asymptotes by finding the zeros of the denomination of the rational function in lowest terms.

The graph of $R$has one vertical asymptote at $x=1$.

There is no horizontal asymptote.

So, let us calculate the oblique asymptote. This can be found by dividing the numerator by denominator and the part except the remainder is the asymptote.

In $R$when we do this procedure, we can find,

$y=\frac{2x+4}{x-1}$

The polynomial has a horizontal asymptote at$y=2$.

Step 3 Now plot a graph by using the known values.