This section covers:

**Revisiting Direct and Inverse Variation****Polynomial Long Division****Asymptotes of Rationals****Drawing Rational Graphs — General Rules****Rational Inequalities, including Absolute Values****Applications of Rational Functions****More Practice**

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 8*x*), 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”.) We will learn later that asymptotes are examples of **limits**; meaning that something gets closer and closer to a number, without actually touching it.

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

**gets larger and larger,**

*x***gets closer and closer to 0. This is because as**

*y***gets larger and larger, we’ll get a smaller and smaller fraction (convince yourself by putting in your calculator), but**

*x***can never**

*x***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,

*goes towards 0?) Horizontal asymptotes are also called*

**x****end behavior asymptotes**, since they occur when

**gets very small and also very big.**

*x*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

*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*

**x***moves graphs up and down).*

**y**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

And more:

The way I like to remember the **horizontal asymptotes (HAs)** is: **BOBO BOTN EATS DC** (**B**igger **O**n **B**ottom, asymptote is **0**, **B**igger **O**n **T**op, **N**o asymptote, **E**xponents **A**re **T**he **S**ame, **D**ivide **C**oefficients).

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 thevalue of the hole, use the**y****crossed out version**and plug in thevalue. You know that part of the curve of the graph goes close to that point, but you have to graph a small circle there.*x*- Draw any
**VA asymptotes**(**non-removable discontinuities**) 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
(where*x**i*ntercepts*y*= 0), and(where*y*intercepts*x*= 0). - (More Advanced) See if
**the function crosses any horizontal asymptotes**by setting the original function equal to the HA. Solve for; you already have the*x*(from the asymptote).*y* **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.- Have your graphs “
**hug every asymptote”**, but remember that you will never have more than one point on a vertical line, since we’re drawing functions.

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. Or sometimes you see cubic looking curves inside the VA’s with graphs in opposite corners.

Also note that if any factors on the bottom are repeated, they are said to have a **multiplicity**. For factors (asymptotes) with an **odd multiplicity** (odd exponent), the graph will alternate directions on either side of the **VA**; for **even multiplicity **(even exponent), the graph will be in the same direction on both sides of the **VA**.

With **slant (oblique) asymptotes**, the curves will slant. And always check points to make sure you’ve graphed correctly!

**Examples:**

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): 🙄

You may be asked to look at a rational function graph and** find a possible equation for the rational function graph**:

# 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

It’s not too bad to see **inequalities of rationals from a graph**. Look at this graph to see where . Notice that we have** ranges of x values** in the two cases:

## Solving Rational Inequalities Algebraically Using a Sign Chart

The easiest way to solve rational inequalities algebraically is using the **sign chart method**, which we saw here in the** Quadratic Inequalities Section Method.**

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:

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

**is negative. Notice also how we had to use the**

*x***Quadratic Formula**to get the critical points when

**is negative:**

*x*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:

**Problem:**

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?

**Solution:**

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

**values. You can see that the maximum concentration of**

*y***2.5mg**occurs after

**1 hour**:

**Learn these rules and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Graphing and Finding Roots of Polynomial Functions** — you are ready!

This helped me a lot, and I’m usually really good at math, especially algebra. it’s one of my favorite subjects, which is part of the reason I acted upon my feelings of uneasiness upon this subject of rational functions. Thank you so much. It was truly such a great, help.

P.S. I really liked the BOBO BATN EATN DC. Bigger On Bottom, asymptote is 0, Bigger On Top, No asymptote, Exponents Are The Same, Divide Coefficients

Typo. Under “Asymptotes of Rationals,” end of 4th paragraph, ” … y moves graphs up and own).”

*down*

: )

Thank you!!! Let me know if you find any more!!! Lisa

Glad to help. Are the GDC pics new? I don’t remember them from before. Keep up the great work! My students know your site is *my preferred reference* for them, and I often link it to our course moodle page! Please do complete the Calculus section before August! Thanks, and best to you, Lisa.

I did a search for “asymptote chart” and this pulled up. It is like a light bulb suddenly lit up for me! Going though math problems, the information was all mixed up; listing all asymptotes, X Yintercepts, which ones have oblique, I was so confused. Your chart helped break everything down visually in one place so I could actually understand. You may never know how grateful I am for finding this page!!! A million thank yous!

Thank you SO MUCH for taking the time out to write me! I so appreciate this; this is what makes me keep writing! I’m glad you got some useful information, and please spread the word about She Loves Math! Lisa

Hi I’m wondering what to do when there is both a hole and a VA?

I had a homework question of (x+2)(x+5) over (x+5)^2(x-6)

In this case the (x+5) would cancel out on the top and bottom but still have one (x+5) on the bottom

Would -5 be considered a VA? and how would the graph look like

Thank you so much as I can’t seem to find the answer to my question anywhere else!

That’s a great question! I graphed this on a graphing calculator and it looks like it’s still a vertical asymptote, even after canceling out the “hole”. Does that make sense? Lisa

Thank you so much for your help!

Thank you for your help and I would also recommend you add quadratic asymptotes and the direction of the graphs based on the VA multiplicity.

Plano ISD just changed their pre-cal (as well as alg2 and alg1) curriculum so I’m not sure if you have had to encounter this in the past 🙂

Thank you so much for pointing this out!! I will add the vertical asymptote trick soon. And please let me know what else should be added!! Thanks again, Lisa

This is a question a student asked me:

is (x^2-x-12)/(x+3) a polynomial? After the numerator is factored.

I’ve never been asked that question so I’m not sure how to answer it. My gut says that when we names a function we name the original equation. So it wouldn’t be a polynomial.

It would be if we didn’t see the original function or if the simplified form is what we are given. Thoughts anyone? This question came from an 8th grader.

Thanks for writing! This isn’t a polynomial, since a polynomial must have a domain of all real numbers, and this function has a removable discontinuity at x = -3. Does this make sense? Lisa

Thank you! I didn’t know that part of the definition. What about x=y^2?

Thanks for writing! Are you talking about the definition of a rational function? x = y^2 is actually not a function at all, since it doesn’t pass the vertical line test. Lisa