# Find all the vertical and horizontal asymptotes, maximum and minimum, and points of inflection of...

## Question:

Find all the vertical and horizontal asymptotes, maximum and minimum, and points of inflection of the graph and sketch the graph.

{eq}y = \frac{x+2}{x^2+2x} {/eq}

## Graphing Rational Functions:

Given a rational function {eq}R(x) = \frac{p(x)}{q(x)}{/eq} where {eq}p(x) = a_1x^n + a_2x^{n-1} + \cdot\cdot\cdot {/eq} and {eq}q(x) = b_1x^m + b_2x^{m-1} + \cdot\cdot\cdot{/eq} are polynomial functions of {eq}x{/eq}.

Vertical Asymptotes

The vertical asymptotes of {eq}R{/eq} are the zeroes of the denominator {eq}q(x){/eq}. However, if {eq}x=x_0{/eq} is both a zero of {eq}p(x){/eq} and {eq}q(x){/eq}, then {eq}R{/eq} has a hole at {eq}x=x_0{/eq}.

Horizontal Asymptote

Comparing the degrees {eq}n {/eq} and {eq}m{/eq} of {eq}p(x){/eq} and {eq}q(x){/eq}, respectively. If:

• {eq}n < m{/eq}, then the x-axis is the horizontal asymptote of the rational function;
• {eq}n=m{/eq}, then the line {eq}y=\frac{a}{b}{/eq} is the horizontal asymptote of the rational function; or
• {eq}n>m{/eq}, then there is no horizontal asymptote but an oblique asymptote to the graph of the rational function.

Maximum, Minimum and Inflection Points

The first-order derivative {eq}R'(x){/eq} of the rational function gives the slope of the function as well as the location of the local extremes;

while the second-order derivative {eq}R''(x){/eq} determines its concavity as well as the location of the inflection points and if the local extreme is a local maximum or minimum.

Specifically, given {eq}x = a{/eq}, if:

• {eq}R'(a)=0{/eq} and {eq}R''(a)>0{/eq}, then {eq}x=a{/eq} is a relative minimum;
• {eq}R'(a)=0{/eq} and {eq}R''(a)<0{/eq}, then {eq}x=a{/eq} is a relative maximum;
• {eq}R''(a)=0{/eq}, then {eq}x=a{/eq} is an inflection point.

Given the rational function {eq}R(x) = \frac{x+2}{x^2+2x} = \frac{x+2}{x(x+2)}{/eq}.

Vertical Asymptotes

Taking the zeroes of the denominator,

{eq}...

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