Doug C. answered • 4d
Math Tutor with Reputation to make difficult concepts understandable
A rational function looks like f(x) = g(x)/h(x) where g and h are polynomials with no common factor. If they do have a common factor, there is a hole where that common factor is equal to zero.
The fact that there is a vertical asymptote at x = 1, means the denominator has a factor (x - 1).
f(x) = 1/(x -1 ) has a vertical asymptote at x = 1 because f(1) is undefined (cannot divide by zero).
The fact that there is a hole as x = -1 means there is a common factor in the numerator and denominator:
(x + 1).
f(x) = [(x + 1)}/ [(x + 1)(x - 1)]
At this point f(1) and f(-1) are undefined.
Right now the degree of the numerator is less than the degree of the denominator, so the horizontal asymptote is y = 0. We need the degree of the numerator and denominator to be equal. We can do this by adding an x factor to the numerator. And we need to add a constant factor as coefficient of x to make the horizontal asymptote be y = -2. That constant will be -2. This means that as x -> ±∞, the y-values of the function will go towards -2.
f(x) = [(-2x)(x+1)]/[(x+1)(x - 1)]
Watch what happens as x gets really big (ignoring the x + 1 factors which cancel)
f(1000) = -2000/999 ≈ -2.002
f(0) = 0, so the y-intercept is still not (0,3). Instead of just the "x" factor, let's change that to a binomial. Note that when x = 0, the denominator has a value of -1, which means we want the numerator to equal -3 when x = 0.
-2(0 + k) = -3
k = 3/2
f(x) = [-2(x + 3/2)(x+1)]/[(x+1)(x - 1)]
If you want to, distribute the -2:
f(x) = [(-2x - 3)(x + 1)]/(x + 1)(x - 1)]
f(0) = -3/-1 = 3
Cancelling the (x + 1) factors leaves:
g(x) = (-2x-3)/ (x - 1)
g(-1) = -1/-2 = 1/2
So the hole for f(x) is located at (-1, 1/2).
Take a look at this graph to see all of the above (vertical and horizontal asymptotes, y-intercept, hole)
desmos.com/calculator/8dc2bohc0x
