How do you graph $y=(x+2)/(x+3)$ using asymptotes, intercepts, end behavior?

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How do you graph $y=(x+2)/(x+3)$ using asymptotes, intercepts, end behavior?

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Answer 1 · 2016-12-20T12:21:38

graph{(x+2)/(x+3) [-10, 10, -5, 5]}

$y$ -intercept is $2/3$

$x$ -intercept is $-2$

vertical asymptote at $x=-3$

$y<1$ when $x> -3$

$y>1$ when $x< -3$

Explanation:

asymptote:

$n/0$ = undefined

$therefore$ if $x+3 = 0$ , $y$ is undefined.

this means that it is not on the graph, and so is shown as the asymptote.

when $x+3 = 0$ , $x = 0-3$

$x = -3$

intercepts:

$y$ :

the $y$ -intercept is when $x = 0$

$y = (x+2)/(x+3)$

$y = 2/3$

$y$ -intercept is $2/3$

$x$ :

the $x$ -intercept is when $y=0$

$(x+2)/(x+3)=0$

the numerator has to be $0$ , since $0/n = 0$

this means that $x+2=0$

when $x+2 = 0$ , $x = 0-2$

$x$ -intercept is $-2$

end behaviour:

$y<1$

$x+2$ is always less than $x+3$ . with both positive and negative numbers.

the fraction $(x+2)/(x+3)$ cannot be simplified to $(>=1)/1$ if $x> -3$

however, it is vice versa for negative numbers, since smaller negative numbers have a higher absolute value (distance from $0$ ),

e.g. $-3/-2 = |3|/|2| = 3/2$

this means that $(x+2)/(x+3)$ cannot be simplified to $(<=1)/1$ if $x< -3$

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How do you graph $y=(x+2)/(x+3)$ using asymptotes, intercepts, end behavior?

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