Question

Determine the holes, vertical asymptotes and horizontal asymptotes of the

rational function


Could you also show your work because I need to know how to do this for an upcoming test. Thanks in advance.

Answer 1

I could give you some answers and then depend on you to ask questions until you're satisfied that you know enough to take your upcoming test.

horiz. asymp.: Focus on the dominant (highest x power) terms of the numerator and denominator. They are 3x^2 and x^2 repsectively. Dividing 3x^2 by x^2, we get 3. The horiz. asympt. is merely y=3. It's not always this easy, tho'.

vert. asymptote depends upon the denominator alone. You must find the roots of the denom.: x^2 + 7x + 12. They are -3 and -4. Notice how x^2 + 7x + 12 = (x+3)(x+4). The vertical asymptotes are x=-3 and x=-4.

Now just suppose that the numerator and denominator of your function share a common factor (which means you can cancel out that factor). But we're not off the hook yet! Suppose that the factor common to numerator and denominator is (x+3). You can cancel this out, removing it from both numerator and denominator, but when you graph the rational function, you must place a circle at the root showing that x cannot have that particular value. If, as I supposed, (x+3) is a common factor, then leave a circle at x=-3 on your graph. "Holes" show up in straight lines. For more info on holes I urge you to do an int. search for "holes in rational function graphs" ... I found a great explanation that way.