This section covers Revisiting Direct and Inverse Variation, Polynomial Long Division, Asymptotes of Rationals, Drawing Rational Graphs — General Rules, Rational Inequalities, including Absolute Values, and Applications of Rational Functions.
Now that we know how to work with both rationals and polynomials, we’ll work on more advanced solving and graphing with them.
Revisiting Direct and Inverse Variation
We went over Direct, Inverse, Joint and Combined Variation here, but little did we know that we were working with Rational Functions when we were solving Inverse or Combined variation problem! This is because we had variables in the denominators for these types of problems.
Polynomial Long Division
We need to take a minute (ha ha) and talk about long division with polynomials. Long division with polynomials is sometimes needed when the degree (highest exponent in any variable) in the numerator is larger than the degree of the denominator. If the denominator is just one term (a monomial like 8x), we just put each term in the numerator over the denominator. This is also called simplifying or reducing the fraction:
When there are more than two terms on the bottom, it gets a little more complicated, and we have to do polynomial long division. There’s actually an easier way to do this with Synthetic Division, which we’ll learn about later, but let’s work with long division first. It’s really cool, since we can divide polynomials very similar to “regular” numbers. Notice how the steps line up:
Let’s do more polynomial long division. Notice if we are missing a term in the dividend part (under the division sign), we have to create one with a coefficient of 0, just so we can line up things when we do the dividing.
Asymptotes of Rationals
Because rationals typically have variables in the denominator, graphing them can be a bit tricky. We’ll introduce here the notion of an asymptote, or a graph that gets closer and closer to a line but never hits it. (It comes from a Greek word, meaning “not falling together”.)
The reason we see asymptotes in rationals is because, again, there are typically x values (domains) where the function or graph does not exist at all, since we can’t divide by “0”. One of the simplest rational functions, the inverse function, is , as shown on the left below. (We saw this in the Parent Functions and Transformations Section here). Notice how, as x gets larger and larger, y gets closer and closer to 0. This is because as x gets larger and larger, we’ll get a smaller and smaller fraction (convince yourself by putting in your calculator), but x can never be 0, since we can’t have a 0 in the denominator.
In this graph, we have a horizontal asymptote at “y = 0” and a vertical asymptote at “x = 0”. (Do you see how this asymptote is vertical: as y gets very small and very large, x goes towards 0?) Horizontal asymptotes are also called end behavior asymptotes, since they occur when x gets very small and also very big.
I like to call (0, 0) the “anchor point” of the graph, since it’s the point where the two asymptotes intersect. Vertical asymptotes are sometimes written as VA, and end behavior asymptotes are written as EBA. Again, end behavior asymptotes are called such since they exist at the extreme areas of the x: where . Horizontal asymptotes (also written as HA) are a special type of end behavior asymptotes. The graph on the right shows what happen when we shift the graph of “2 units to the right, and 3 units up”. (We learned about transformations of functions in the Parent Functions and Transformations section here: anything we add or subtract to the x part of the equation moves graphs back and forth and in the opposite way you’d think, and anything you add or subtract to the y moves graphs up and down).
So the “anchor point” of the graph to the right is (2, 3), and the asymptotes are x = 2 (VA) and y = 3 (EBA/HA).
Here are more inverse function graphs that you may have to draw and also shift or transform:
Let’s try graphing a few more generic inverse functions, just for fun:
Continuous Versus Discontinuous Functions
We talked about continuous versus discontinuous (or non-continuous) functions in the Piecewise Functions Section here, but notice that the functions above are discontinuous, meaning that you have to lift up your pencil when you draw them from left to right. Continuous Functions are just that; they could theoretically be drawn from “left to right” (or less commonly “down to up”) without picking up a pencil. So any linear function, for example, is continuous. Some rational functions are also continuous, as we’ll see later.
Drawing Rational Graphs – General Rules
We can actually look at more complicated forms of rational functions and, from just a small set of rules, roughly draw the graph of that function – it’s like magic ;)! We may need a T-chart to help us out, but we’ll be able to graph most rational functions pretty quickly.
The table below shows rules and examples. You’ll find that these same rules apply to the graphs above (after finding a common denominator and combining terms if necessary), but usually you are taught those graphs separately.
Remember that the degree of the polynomial is the highest exponent of any of the terms, and if the polynomial is in factored form, you have to multiply the variables (or add the exponents and find the highest exponent) to find the degree. For example, the degree of and the degree of
The way I like to remember the horizontal asymptotes (HAs) is: BOBO BOTN EATS DC (Bigger On Bottom, asymptote is 0, Bigger On Top, No asymptote, Exponents Are The Same, Divide Coefficients).
Note that there can be multiple vertical asymptotes, but only one EBA (HA or slant/oblique) asymptote. Note that also the function can intersect the EBA asymptote, but not intercept the vertical asymptote(s). Also, sometimes the function intersects the EBA and then come back up or down to get closer to the asymptote.
To create rational graphs without a calculator:
- Factor to see if any removable discontinuities (or holes) exist; cross out on top and bottom. To get the y value of the hole, you can use the crossed out version and plug in the x value. You know that part of the curve of the graph goes close to that point, but you have to graph a small circle there.
- Draw any VA asymptotes from setting anything left in the denominator to 0. (You may get none, but there can be more than one.)
- Draw any EBA (HA or oblique) asymptotes from BOBO BOTN EATS DC and oblique/slant asymptote instructions above. (You may get none, but there will be at most one).
- Determine the x intercepts (where y = 0), and y intercepts (where x = 0).
- (More Advanced) See if the function crosses any horizontal asymptotes by setting the original function equal to the HA. Solve for x; you already have the y (from the asymptote).
- Draw “T charts” to fill in extra “key” points, for example, on the sides of the EBA asymptotes.
- Domain is everything except where the removable discontinuities or asymptotes exist.
- Remember that with EBA and vertical asymptotes, the graphs usually are in opposite corners; for example, if one curve is in the upper right part of the asymptotes, the other curve could be in the lower left. If you have two or more vertical asymptotes, the graph can look like a parabola inside the two asymptotes with outside curves on either the top or bottom (opposite the “parabola”). Or sometimes you see cubic looking curves inside the VA’s with graphs in opposite corners. With slant (oblique) asymptotes, the curves will slant. But always check points to make sure!
Here are more advanced examples, with a slant (oblique) asymptote and a pass-through asymptote:
Note that if you end up with a line after taking out the removable discontinuity, the EBA is actually considered that line (could be a trick test question):
Rational Inequalities, including Absolute Values
Solving rational inequalities are a little more complicated since we are typically multiplying or dividing by variables, and we don’t know whether these are positive or negative. Remember that we have to change the direction of the inequality when we multiply or divide by negative numbers. So when we solve these rational inequalities, our answers will typically be a range of numbers.
Rational Inequalities from a Graph
Solving Rational Inequalities Algebraically Using a Sign Chart
A sign chart or sign pattern is simply a number line that is separated into partitions (or intervals or regions), with boundary points (called “critical values“) that you get by setting the factors of the rational function (both in numerator and denominator) to 0 and solving for x.
Sign charts are easy and a lot of fun since you can pick any point in between the critical values, and see if the whole function is positive or negative. Then you just pick that interval (or intervals) by looking at the inequality. Generally, if the inequality includes the = sign, you have a closed bracket, and if it doesn’t, you have an open bracket. But any factor that’s in the denominator must have an open bracket (for the values that make it 0), since you can’t have 0 in the denominator.
The first thing you have to do is get everything on the left side (if it isn’t already there) and 0 on the right side, since we can see what intervals make the inequality true. We have to have only one term on the left side, so sometimes we have to find a common denominator and combine terms.
You can always use your graphing calculator to check your answers, too. Put in both sides of the inequalities and check the zeros, and make sure your ranges are correct!
Also, it’s a good idea to put open or closed circles on the critical values to remind ourselves if we have inclusive points (inequalities with equal signs, such as and ) or exclusive points (inequalities without equal signs, or factors in the denominators).
Let’s do some examples:
And here’s another one where we have to do a lot of “organizing” first, including moving terms to one side, and then factoring:
Here’s one more where we can ignore a factor that can never be 0 (even though it’s technically not a rational function, but a polynomial function):
Sometimes we get a funny interval with a single x value as part of the interval:
And here’s one where we have a removable discontinuity in the rational inequality, so we have to make sure we skip over that point:
Here are a couple that involves solving radical inequality with absolute values. (You might want to review Algebraic Equations with Absolute Value before continuing on to this topic.)
Let’s do a simple one first, where we can handle the absolute value just like a factor, but when we do the checking, we’ll take into account that it is an absolute value. Note that we can do this since the absolute value doesn’t have to be multiplied by other variables to solve the inequality.
Here are more complicated ones, where the absolute value may need to be multiplied by other variables (think of if you had to cross multiply). Notice how it’s best to separate the inequality into two separate inequalities: one case when x is positive, and the other when x is negative. Notice also how we had to use the Quadratic Formula to get the critical points when x is negative:
Here’s one more that’s a bit tricky, since we have two expressions with absolute value in it. In this case, we have to separate in four cases, just to be sure we cover all the possibilities.
There’s one more that we did here in the Compositions of Functions, Even and Odd, and Increasing and Decreasing section, when we worked on domains of composites.
Applications of Rational Functions
We did some of application earlier here in the Rational Expressions and Functions section, but here’s another that deals more with the concept of asymptotes:
The concentration of a drug is monitored in the bloodstream of a patient. The drug’s concentration can be modeled by where t is in hours, and is in mg.
a) What is the equation of the horizontal asymptote associated with this function? What does it mean about the drug’s concentration in the patient’s bloodstream as time increases?
b) When does the maximum concentration of the drug occur, and what is this maximum concentration?
a) The asymptote of the function is y = 0, since the degree on the bottom is greater than the degree on the top. So what this means is that, as time goes on, the drug is basically negligible in the patient; its concentration gets closer and closer to 0 mg.
b) To find the maximum concentration, let’s put the equation in the graphing calculator and use the maximum function to find both the x and y values. You can see that the maximum concentration of 2.5mg occurs after 1 hour:
Learn these rules and practice, practice, practice!
On to Graphing and Finding Roots of Polynomial Functions — you are ready!