Rational functions are defined as the quotient of two polynomials.† With a basic understanding of polynomials, one can see there are many possible explorations regarding various degrees and quotients.†
First, letís consider the quotient of two linear functions.†
I will define †and.
The most interesting portion of this graph is the presence of a vertical asymptote at x =4.† The asymptote exists here because the graph is undefined when x =4 and the graph will bend away from this number which makes the denominator zero.† Furthermore, there is a horizontal asymptote at y =1 because the degree of the numerator is the same as the degree of the denominator.†
It is also interesting to note that the x-intercept of the quotient of f and g, from here on referred to as h(x), is x=-7.† This is precisely the same as the x-intercept for f.† Perhaps we should verify this with another quotient of linear functions to see whether it holds true.† It should, seeing that the values which make y =0 are the same for f and h.†
As you can see, it holds true.† Through basic algebraic manipulation, we can see that it will always be true.†
Now, letís consider the quotient of two quadratic functions.† Suppose †and .†† The quotient of these two functions is interesting because there is a vertical asymptote, a hole, and a horizontal asymptote.†
Through factoring and other algebraic manipulation, we can see there is a vertical asymptote at x =3 and a hole at x =2 (the coordinate (2,-4) since it is very difficult to see!).† It may not be as obvious that there is also a horizontal asymptote at y =1.† In order to find x and y-intercepts, we simply need to substitute x =0 and y =0.† It turns out that the x-intercept is the same for f(x) and h(x) like it has been in every example thus far.† This is because y =0 on the x-axis and the only way for this to be true is if the numerator (or f(x)) is equal to zero.† If the denominator (or g(x)) is equal to zero, then we will have an undefined function at this location (also known as the asymptote).†
Now we can again consider the quotient of two quadratic functions.† Let †and .†† These graphs are particularly interesting because f(x) and g(x) only have complex roots.† Their quotient does not behave in a similar way as the examples above for this reason.† This rational function lacks true asymptotes and holes because there is not a place where the graph is undefined in the real number system.† Instead we see an interesting result of no vertical asymptotes.† However, we do see a horizontal asymptote at y =1.† It is also interesting to note that the y-intercept is the same for g(x) and h(x) (y =3).†
What about the quotient of two cubic functions?
According to the pattern we have seen so far, we can predict a horizontal asymptote at y =1 (since we know the degrees of the numerator and denominator will be the same).† We can also predict that f(x) and h(x) will have the same x-intercepts.† The presence of a hole or a vertical asymptote is dependent on the numeric values of each function. Letís consider †and .† Let their quotient be h(x).†
Our predictions are correct.† Firstly, we have a horizontal asymptote at y =1 (again, because the degrees are the same, we take the ratio of the leading coefficients to find the horizontal asymptote).† We also see the x-intercepts are the same for f(x) and h(x).† More specifically, we see vertical asymptotes at x =1 and x =-6.† In addition, we should also see a hole at x =0.†
What about the quotient of two quartic functions?
More than likely, we will see the same pattern with quartic polynomials as with cubics, quadratics, and linear.† We should have the same horizontal asymptote arising from the ratio of the leading coefficients since the degrees will be the same, and f(x) and h(x) should have the same x-intercepts.† Letís verify this is true with an example.†
Let †and .† Again, allow h(x) to represent the quotient of these two polynomials.†
This is an interesting graph because the horizontal asymptote is not as clear in this example.† Much like the other rational functions, this one has a vertical asymptote at x =2 and contains a hole at x =0.† Furthermore, f(x) and h(x) have the same x-intercepts (x =-1 and x =4).†
So far, we have only investigated the quotient of functions of the same degree.† What happens if the degree of the function in the numerator is higher than the degree of the denominator?†
Letís suppose †and .
Again, we see that f(x) and h(x) have the same x-intercepts and the vertical asymptotes are determined by g(x) =0.† The horizontal asymptote is missing from this picture- which is exactly what one should expect if the numerator degree is greater than the denominatorís degree.†
Perhaps we should also consider the quotient of polynomials in which the degree of the numerator is less than the degree of the denominator.†
Suppose we take this time.
We see a horizontal asymptote at y =0, which is what one should expect when the degree of the denominator is larger than the degree of the numerator.† We again see that h(x) and g(x) have the same x-intercepts.†
Not all asymptotes are vertical or horizontal lines.† The majority of asymptotes students encounter are, but we also have ďslant asymptotesĒ.† A slant asymptote is exactly like what it sounds- an asymptote which is slightly tilted rather than a vertical or horizontal line.† Slant asymptotes are essentially the resultant polynomial or the quotient of the two polynomials.† Furthermore, slant asymptotes only result if the degree in the numerator is greater than the degree of the denominator since there cannot be a horizontal asymptote.
For instance, letís consider the rational function .
As you can see in the picture, it appears that there is some sort of asymptote guiding the path of this graph.† To find this slant asymptote, we must use long division (or synthetic division if possible).†
So, our quotient is the equation of the slant asymptote: .