Question

Graph the rational function.=fx+2x6+x4Start by drawing the vertical and horizontal asymptotes. Then plot two points on each piece of the graph. Finally, click on the graph-a-function button.

Answer 1

Given the function:

$f(x)=\frac{2x+6}{x+4}$

Let's graph the function.

To graph, let's first find the vertical and horizontal asymptotes.

To find the vertical asymptote, let's equate the denominator to 0 and solve for x.

$\begin{gathered} x+4=0 \\ \text{ Subtract 4 from both sides:} \\ x+4-4=0-4 \\ x=-4 \end{gathered}$

The vertical asymptote is:

x = -4

To find the horizontal asymptote, apply the condition:

n = m

Where n is the degree of the numerator while m is the degree of the denominator.

Since n = m, the horizontal asymptote will be:

$y=\frac{a}{b}$

Where:

a = 2

b = 1

$\begin{gathered} y=\frac{2}{1} \\ \\ y=2 \end{gathered}$

The horizontal asymptote is:

y = 2

We have a sketch of the asymptotes below:

Now, let's find two points each.

We have the following:

$\begin{gathered} When\text{ x =1: }f(2)=\frac{2(0)+6}{0+4}=\frac{6}{4}=1.5 \\ \\ When\text{ x = -3: }f(3)=\frac{2(-3)+6}{-3+4}=\frac{-6+6}{1}=0 \end{gathered}$

We have the points: (1, 1.5), (-3, 0)

Also find two points on the upper part:

$\begin{gathered} When\text{ x =-5: f\lparen}-5)=\frac{2(-5)+6}{-5+4}=\frac{-10+6}{-1}=4 \\ \\ When\text{ x = -6: }f(-6)=\frac{2(-6)+6}{-6+4}=\frac{-12+6}{-2}=\frac{-6}{-2}=3 \end{gathered}$

We have the points: (-5, 4) and (-6, 3)

Plot the points and sketch the graph.

We have the graph of the rational function below:

ANSWER:

Vertical asymptote: x = -4

Horizontal asymptote: y = 2