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.
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)
- If n < m, y="0
- 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.
- If n > m, there is no horizontal 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.