Question

How do you graph `y=(x-5)/(x+1)` using asymptotes, intercepts, end behavior?

Answer 1

Asymptotes: `x=-1` and `y=1`
Intercepts: `(0,-5)` and `(5,0)`
End behavior: As `x -> oo` and `x -> -oo`, `y ->1`

Explanation:

For the asymptote, we have vertical ones and horizontal ones.

For vertical ones, we look at the denominator and determine when it would be `0`. Whenever the denominator is `0`, that is where a vertical asymptote will be, except for when it is a hole.

Horizontal ones are determined by finding the ratio of the coefficient in front of highest exponential value of the numerator and denominator. The highest exponential value on the numerator is x, and its coefficient is 1. The highest exponential value on the denominator is x as well, with its coefficient being 1. So the horizontal asymptote would be `1/1` which is `1`.

The y intercept is when x is zero. So the y intercept would be `(0,-5)`. You can get this by plugging x into the equation.

The x intercept is when y is zero. When `x=5`, then y would be 0, so the x intercept would be `(5,0)`.

You can determine the end behavior just by looking at the horizontal asymptote. The horizontal asymptote tells you where the y value is going towards, which is `x=1` in this case.