Graphing Rational Functions

Question

x^2 -3x +10

y= --------------------

x+4

All I am trying to do is find the asymptote lines for this equation.

I can see that: x= - 4, but am having trouble solving for y.

I am told the answer is y = x-7, but am not sure why.

Should I write this as:

(x-5) (x+2)

y= -------------

x+4

If yes, how do I solve?

If no, ??

Answer 1

Whenever the power of the numerator is one greater than the power of the denominator, we have a slant asmptote, which is what y = x - 7 is. We can use long-division or a version of it, synthetic division. to find the asymptote.

Using long-division

x -7 r 38

x+4 | x^2 -3x + 10

x^2 4x

-7x + 10

-7x -28

38

Using synthetic division (much easier to write)

-4 | 1 -3 10

-4 28

1 -7 38

y = x - 7 + 38/x-4

As x goes to infinity and -infinity, the last term goes to zero, leaving us with y = x - 7

Thus, y = x - 7 is the slant asymptote.

As a check, note that (x-7)(x+4) = x² -3x -28