Rational Functions

Question

find the vertical asymptotes, horizontal asymptotes, slant asymptote, x-intercept(s), y-intercept, if applicable, of the functions. 

2x²-2x-12/x²-9

Answer 1

The numerator factors as 2(x-3)(x+2)/(x-3)(x+3)

Note x-3 in the numerator and denominator. There will be a hole there.

There will be a vertical asymptote at x+3 = 0, or x = -3

When the leading powers of the numerator and denominator are the same, there will be a horizontal asymptote at the ratio of the leading coefficients unless the fraction reduces to a constant.

In this case, the leading powers are 2 in num. and den. , and their ratio is 2/1, so the horizontal asymptote is at 2.

There isn't a slant asymptote.

The x intercept is where the numerator equals 0 unless the denominator also equals 0 there.

x+2 = 0; x = -2

The y-intercept is when x equals 0; -12/-9 = 4/3

Answer 2

For

2x² - 2x -12/(x² - 9)

Factor the numerator and denominator

(2x + 4)(x - 3)/(x + 3)(x -3)

Notice the factor (x -3) cancels out of the numerator and denominator leaving

(2x + 4)/(x +3)

Since the denominator cannot be zero then

x + 3

cannot be zero

so x cannot be negative 3

So negative 3 would be a vertical asymptote

The x intercepts can be found by setting the fraction equal to zero

(2x + 4)/(x +3) = 0

Multiply both sides of the equation by the denominator, this leaves

2x + 4 = 0

x = -2

So the x intercept is (-2,0)

The y intercept is found by setting x equal to zero in the original fraction

2x² - 2x -12/(x² - 9)

2(0²) - 2(0) - 12/(0² - 9)

This leaves

-12/-9 = 4/3

So the y intercept is (0, 4/3)

You can check for the horizontal asymptote by taking the limit of your function or by graphing it. There is a value that exits as the function approaches +/- infinity

Finally a slant asymptote occurs when the degree of the polynomial in the numerator is higher than the degree of the polynomial in the denominator Check your original fraction to see if this situation exists.

You can check the numbers, and look for the asymptotes by graphing your original fraction or your simplified fraction

I'll leave it to you to apply the rules and find the horizontal and slant asymptotes if they exist.

I hope this helps