To graph the given rational function, we will find its domain, asymptotes, and intercepts. Then, we will find points using a table of values. Finally, we will plot and connect those points.

Domain

Consider the given function.

y = \frac{3 x}{( x + 2 ) ^{2}}

Recall that division by zero is not defined. Therefore, the rational function is undefined where

(x + 2)^{2} = 0 .

(x + 2)^{2} = 0 \Leftrightarrow x = - 2

This means that

x = - 2

is not included in the domain.

\underline{Domain} All real numbers except x = - 2

Asymptotes

Asymptotes can be vertical or horizontal lines.

Vertical Asymptotes

Once again, let's consider the given function.

y = \frac{3 x}{( x + 2 ) ^{2}}

Note that we cannot cancel out common factors. Therefore, there are no holes. Also, if the real number

a

is not included in the domain, there is a vertical asymptote at

x = a .

In this case, we have a vertical asymptote at

x = - 2 .

Horizontal Asymptotes

To find the horizontal asymptotes, we can use a set of rules. To properly use these rules, in the following table $m$ and $n$ must be the highest degree of the numerator and denominator.

$y = \frac{a x ^{m}}{b x ^{n}}$	Asymptote
$m < n$	$y = 0$
$m > n$	none
$m = n$	$y = \frac{a}{b}$

We must expand the denominator of the function.

y = \frac{3 x}{( x + 2 ) ^{2}}

ExpandPosPerfectSquare

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$

y = \frac{3 x}{x ^{2} + 4 x + 4}

Let's look at the degrees of the numerator and denominator for our function.

y = \frac{3 x ^{1}}{x ^{2} + 4 x + 4}

We can see that the degree of the denominator is higher than the degree of the numerator. Therefore, the line

y = 0

is a horizontal asymptote.

Intercepts

The intercepts of the function are the points at which the graph intersects the axes.

$x -$ intercepts

The

x -

intercepts are the points where the graph intersects the

x -

axis. At these points, the value of the

y -

coordinate is zero. Let's substitute

0

for

y

in the given function and solve for

x .

y = \frac{3 x}{( x + 2 ) ^{2}}

Substitute

$y = 0$

0 = \frac{3 x}{( x + 2 ) ^{2}}

▼

Solve for

x

MultEqn

$LHS \cdot (x + 2)^{2} = RHS \cdot (x + 2)^{2}$

0 = 3 x

DivEqn

$LHS / 3 = RHS / 3$

0 = x

RearrangeEqn

Rearrange equation

x = 0

There is an

x -

intercept at

(0, 0) .

$y -$ intercept

The

y -

intercept is the point where the graph intersects the

y -

axis. At this point, the value of the

x -

coordinate is zero. Let's substitute

0

for

x

in the given function and solve for

y .

y = \frac{3 x}{( x + 2 ) ^{2}}

Substitute

$x = 0$

y = \frac{3 ( 0 )}{( 0 + 2 ) ^{2}}

▼

Solve for

y

ZeroPropMult

Zero Property of Multiplication

y = \frac{0}{( 0 + 2 ) ^{2}}

IdPropAdd

Identity Property of Addition

y = \frac{0}{2 ^{2}}

CalcPow

Calculate power

y = \frac{0}{4}

CalcQuot

Calculate quotient

y = 0

There is a

y -

intercept at

(0, 0) .

Graph

Let's make a table of values to graph the given function. Make sure to only use values included in the domain of the function.

$x$	$\frac{3 x}{( x + 2 ) ^{2}}$	$y = \frac{3 x}{( x + 2 ) ^{2}}$
$- 5$	$\frac{3 ( - 5 )}{( - 5 + 2 ) ^{2}}$	$\approx - 1.667$
$- 4$	$\frac{3 ( - 4 )}{( - 4 + 2 ) ^{2}}$	$- 3$
$- 3$	$\frac{3 ( - 3 )}{( - 3 + 2 ) ^{2}}$	$- 9$
$- 1$	$\frac{3 ( - 1 )}{( - 1 + 2 ) ^{2}}$	$- 3$
$1$	$\frac{3 ( 1 )}{( 1 + 2 ) ^{2}}$	$\approx 0.333$

Finally, let's plot and connect the points. Do not forget to draw the asymptotes and to plot the intercepts and holes, if any.

Exercise