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)=(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=(a)/(b)`

Where:

a = 2

b = 1

`\begin{gathered} y=(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)=(2(0)+6)/(0+4)=(6)/(4)=1.5 \\ \\ When\text{ x = -3: }f(3)=(2(-3)+6)/(-3+4)=(-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)=(2(-5)+6)/(-5+4)=(-10+6)/(-1)=4 \\ \\ When\text{ x = -6: }f(-6)=(2(-6)+6)/(-6+4)=(-12+6)/(-2)=(-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