Week 3 - Graphing Rational Functions

Here is your weekly checklist.

We've got a pretty good understanding of polynomials now, which really means we have a pretty good understanding of functions in general. Something we've yet to cover, however, are what are known as "discontinuities". Whereas the domain, all the of x values that can be put into a function to get real number outputs, of all polynomials is (-∞,∞), there are other functions that have restrictions to their domains. When it comes to something called a Rational Function, we (usually) come across restrictions to the domain, and the restrictions in this case are called discontinuities. There are several different types of discontinuities that a function can have. Here is a great website that explains each. We introduced limit notation when discussing the end behavior of polynomials, but limit notation can also be used to evaluate the behavior of a function as x approaches any given value (not just positive and negative infinity). You will get into how we use limit notation to identify and define the different kinds of discontinuities in a future precalculus or calculus class, but for now, and when it comes to rational functions, the more useful information about discontinuities involves why they happen and what they look like.

For any function it is generally pretty simple to determine three things: the x-intercept(s), the y-intercept, and the domain. For all functions, if we define y=f(x) (y as a function of x), x-intercepts occur when y=0 [f(x)=0], and y-intercepts occur when x=0 [y=f(0)]. For rational functions, we have discontinuities (restrictions to the domain) when we divide by zero. When a value for x makes the just the denominator of a rational function equal to zero, we end up with a vertical asymptote (also called an infinite discontinuity). When a value for x makes both the numerator and the denominator of a rational function equal zero, we end up with a hole (also called removable discontinuity). Both of these are important to note, and the process for identifying both of them starts by analyzing the denominator of a rational function.

When it comes to graphing rational functions, a great place to start is by first identifying x and y intercepts, then determining the domain (rather, finding restrictions to the domain by setting the denominator equal to zero). We can then classify restrictions to the domain as either a vertical asymptote (more common) or a hole (less common). For rational functions we'll encounter, the final piece of analysis we need to do is to investigate the end behavior, or what is happening to the functions for very large negative values of x, and very large positive values of x.

• Practice problems for basic rational functions: PG 412 #1-5 (concept quiz), #1, 5, 9, 10, 14, 20 (for 14 and 20, just focus on plotting the x and y intercepts, and the vertical asymptote(s). You can then sketch in the end behavior and complete the graph with the information you have. Don't worry about finding where the function is increasing/decreasing or the range.)

By plotting the x and y intercepts, the discontinuities, and examining the end behavior of a rational function, we can "lay out" where the function will be graphed and provide a fairly accurate graph by filling in the blanks with a smooth curve. However, determining the end behavior of a rational function in this manner can not only be tedious, it can be tricky at certain times. For that reason, there is a set of 3 rules that we can refer to that will determine the end behavior of any rational function:

1. If the degree of the numerator of the rational function is less than the degree of the denominator, then f(x) will approach 0 as x approaches ±∞.

2. If the degree of the numerator of the rational function is equal to the degree of the denominator of the rational function, then f(x) will approach the quotient of the leading coefficients as x approaches ±∞. (note: the leading coefficient of a polynomial is the real number that multiplies the term with the highest degree variable).

3. If the degree of the numerator of the rational function is greater than the degree of the denominator of the rational function, then f(x) will approach the quotient of the numerator and denominator as x approaches ±∞.

In this third scenario, Q is what is known as an oblique asymptote. An oblique asymptote differs from a vertical asymptote in that it does not represent a restriction in the domain of f(x) and it is not an infinite discontinuity, but it is similar in that the graph of f(x) will get infinitesimally close to it, but not touch it. Note that when Q is a first degree polynomial (occurs when the degree of the numerator of the rational function is 1 greater than the degree of the denominator) the oblique asymptote is often called a "slant" asymptote. However, when the difference in degree is greater than 1, we can have all sorts of different shapes of oblique asymptotes (you can imagine that a difference of 2 would result in an oblique asymptote in the shape of a parabola).

As before, we can still plot the x and y intercepts, any domain restrictions (infinite or removable discontinuities), and sketch the end behavior of the rational function. In many circumstances we can complete the graph by connecting our information via smooth curves, but sometimes we will need to use a graphing utility (like desmos or a graphing calculator) to get a complete picture of the behavior of the graph.

A graphing utility may need to be used to find relative minimums and maximums, which is information that will be needed to record the range of the function and the intervals over which the function is increasing or decreasing.

• Practice problems for graphing rational functions: Do the three problems from this PDF. Fill in all the requested information in the process of graphing.

Additional materials are posted below.

Once you are ready, take the quiz on Microsoft Forms by Monday, April 20th at 11:59PM. This is mandatory.