3.3 Rational Functions of the form f(x)=(ax+b) / (cx+d)

Today we had a supply and therefore there was no lesson, just a handout. Since we didn't have any verbal explanations of today's lesson, I will just re-write the key points of the handout.

Previously, we had only touched upon reciprocal functions. Basically, that's the inverse of a function that we are used to - for example, x^2-3x+2 is a function that we are used to, and the inverse of this function (or the reciprocal) would be 1/(x^2-3x+2). In a reciprocal function, 1 is divided by a polynomial.

The handout that we received today discussed functions that are also rational, except both the numerator and the denominator of the function are "polynomials in x of degree n and m" (from the sheet).
The general formula for a function of this sort is:
f(x) = (ax^n + b) / (cx^m + d).


We still need to be able to discuss the key points of functions of this nature, including the x- and y- intercepts, as well as both the horizontal and vertical asymptotes. We also need to be able to describe the way the function acts as it approaches the asymptotes from all sides.

• To determine the x-intercept of any rational function, we just set y=0 and solve for x. Similarly, to determine the y-intercept, we set x=0 and solve for y.
• To determine the vertical asymptote of rational functions, we set the denominator equal to 0 and solve for the possible value(s) of x.

Neither of these have changed from reciprocal functions to functions of the form f(x)=(ax+b) / (cx+d) (which we are looking at now). The only difference between the two lies in the way we determine the horizontal asymptote.
There are two methods in determining this. The most simple one comes in the form of general rules: (Remember, n and m are the exponents on the x values in the equation)

1. If n < m, y="0
2. If n = m, the horizontal asymptote is y = (coefficient of the x^n term) / (coefficient of the x^m term) <-- one thing that I got messed up on in the homework is to make sure you look at the COEFFICIENT (the term paired with the x) and not the constant.
3. If n > m, there is no horizontal asymptote.

The second is to create a table of values, using large positive numbers ("close" to positive infinity) and large negative numbers ("close" to negative infinity) and determine the y values, and seeing how close they get to a number without touching (property of an asymptote).

Once you have your asymptotes and intercepts, you can sketch your graph. In terms of which way to draw the graph (end behaviours), I just create a table of values (like the second method for determining the horizontal asymptote) and plot the points that way. There might be an easier way but I guess I haven't found it yet so that works for now.