Question

How do you graph `y=(x^2-7x-60)/(x+3)` using asymptotes, intercepts, end behavior?

Answer 1

See the graph below

Explanation:

As you cannot divide by `0`, `x=-3` is a vertical asymptotes.
As the degree of the numerator `>` the degree of the denominator, we expect a slant asymptote. So we do a long division.
`color(white)(aaa)` `x^2-7x-60` `color(white)(aaaa)` `∣` `x+3`
`color(white)(aaa)` `x^2+3x` `color(white)(aaaaaaaa)` `∣` `x-10`
`color(white)(aaa)` `0-10x-60`
`color(white)(aaaaa)` `-10x-30`
`color(white)(aaaaaaaaa)` `0-30`

`y=(x^2-7x-60)/(x+3)=x-10-30/(x+3)`
So `y=x-10` is an oblique asymptote
`lim_(+-oo)y=lim_(+-oo)x^2/x=+-oo`
When `x=0` `=>` `y=-20`
graph{(y-(x^2-7x-60)/(x+3))(y-x+10)=0 [-83.3, 83.26, -41.65, 41.7]}