How do you graph y=x2+4x5x6 using asymptotes, intercepts, end behavior?

1 Answer
Feb 11, 2018

Asymptotes:

A vertical asymptote occurs when the function is undefined so in this case, when x=6.

We can figure out on which side the y values approach when getting closer to x=6 and on which side the y values approach by plugging in a slightly smaller number than 6 and a slightly bigger number than 6 for x.

If we plug in 6.00001 for x we get a positive number for y and if we plug in 5.9999 for x we get a negative number for y.

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 x=6 there is a slant asymptote.

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 x+10+55x6.

But the slant intercept doesn't include the remainder of the quotient. So there is a slant asymptote at y=x+10

Intercepts

We calculate the intercepts of a rational function by finding the roots of the numerator's function.

So by factoring x2+4x5 we will get the roots of the main function.

These roots are x = -5, 1.

End Behavior

When calculating end behavior you can rewrite the function with only it's leading terms. So our function x2+4x5x6 can be rewritten as x2x which can be simplified to x.

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 x is a large positive number the y will be a large positive number and when the x is a large negative number the y will be a large negative number.