How do you graph #y=(x^2+4x-5)/(x-6)# using asymptotes, intercepts, end behavior?
A vertical asymptote occurs when the function is undefined so in this case, when
We can figure out on which side the
If we plug in 6.00001 for
So directly to the right of the asymptote the values are positive, and directly to the left they are negative.
In addition to the vertical asymptote at
We know this because the magnitude of the numerator's function and the denominator's function differ by +1.
To find where the slant asymptote is, you need to divide the numerator by the denominator. This results in
But the slant intercept doesn't include the remainder of the quotient. So there is a slant asymptote at
We calculate the intercepts of a rational function by finding the roots of the numerator's function.
So by factoring
These roots are x = -5, 1.
When calculating end behavior you can rewrite the function with only it's leading terms. So our function
Now, if we insert a large positive number we would get a large positive number, and similarly if we insert a large negative number we would get a large negative number.
So the end behavior is when the