Question

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

Answer 1

here is the graph:

Explanation:

To graph `y=(x+2)/(x-3)`:

`NPV= 3`
There are no common factors between the numerator and denominator so the vertical asymptote is 3.
`VA=3`

Horizontal Asymptote:
`deg N = degD`

when the numerator and denominator equal the same, you divide the leading coefficients to get the horizontal asymptote.

`HA: y= x/x= 1`

There is no slant asymptote.

Find behaviour near V.A (generally =+/-0.1 away):

at `x=3.1` = `+/+` = positive infinity

at `x=2.9` = `+/-` = negative infinity

so `x=3.1` will start to the right of the vertical asymptote in quadrant I, and `x=2.9` will start to the left of the vertical asymptote in quadrant IV.

The x intercept is the zero of the numerator, so `x=-2`
To find the y intercept, set `x=0` into the equation.

`y=(0+2)/(0-3)`
`y=2/-3` or -0.6667

Join the points smoothly.