The Rational Function Detective

Sections 5.3, 5.4 -- Asymptotic behavior and graphs of rational functions

It costs $5,000 to set up a print run plus $12 per shirt. Print 10 shirts and each one costs you $512. Print 1,000 and each costs $17. Print a million and each costs $12.005. Your average cost keeps dropping, but it will never reach $12. It gets close and stops. That ceiling has a name.

Step 1 of 7: What happens at zero?

Consider $f(x) = \dfrac{1}{x}$. What is $f(0)$?

Step 2 of 7: Values near zero

Still looking at $f(x) = \frac{1}{x}$. What happens as $x$ gets closer and closer to zero?

Compute $f(0.1) = \frac{1}{0.1}$. Type your answer below (within 0.5).

$f(0.1) = $

Step 3 of 7: Values far from zero

Now go the other direction. What happens to $f(x) = \frac{1}{x}$ when $x$ is very large?

Compute $f(100) = \frac{1}{100}$. Type your answer below (within 0.005).

$f(100) = $

Step 4 of 7: A new function

Now consider $f(x) = \dfrac{x - 2}{x - 3}$.

Before graphing, predict: where is the vertical asymptote? (Where does the denominator equal zero?)

Step 5 of 7: The degree rule

The horizontal asymptote depends on how the degrees of the numerator and denominator compare. Three cases, three predictions.

Case 1: $f(x) = \dfrac{3}{x + 1}$. Numerator degree: 0. Denominator degree: 1. What is the horizontal asymptote?

Step 6 of 7: The trap

Now consider $f(x) = \dfrac{(x-2)(x+1)}{(x-2)(x-3)}$.

The denominator is zero at $x = 2$ (because $x - 2 = 0$). Is $x = 2$ a vertical asymptote?

Step 7 of 7: Classify the problems

Given $f(x) = \dfrac{(x-1)(x+2)}{(x-1)(x-4)}$, classify each problem point.

At $x = 1$, the denominator is zero. Is this a hole or a vertical asymptote?

Explore Rational Functions

Build any rational function by adding factors to the numerator and denominator. The graph updates live.

Numerator factors

$(x - r)$, r =

Denominator factors

$(x - r)$, r =

Leading coefficient: 1

$f(x) = 1$

Show asymptotes Show holes Show cancellations

View range: ±10

Add factors above to build a rational function. The graph and analysis will update automatically.

Try This

The T-Shirt Problem

A print shop charges $5,000 to set up a run plus $12 per shirt. The average cost per shirt is:

$C(x) = \dfrac{5000 + 12x}{x}$

What is the horizontal asymptote of $C(x)$?

Medication in the Blood

A patient takes a dose of medication. The blood concentration (in mg/L) after $t$ hours is modeled by:

$C(t) = \dfrac{200t}{t^2 + 25}$

What is the horizontal asymptote of $C(t)$?

Build a Rational Function

Construct a rational function that has all three properties:

A vertical asymptote at $x = -2$
A hole at $x = 4$
A horizontal asymptote at $y = 3$

Leading coefficient: 1

Numerator factors

$(x - r)$, r =

Denominator factors

$(x - r)$, r =

$f(x) = 1$

○ Vertical asymptote at $x = -2$

○ Hole at $x = 4$

○ Horizontal asymptote at $y = 3$

Identify vertical asymptotes by finding where the denominator equals zero (and the numerator does not)
Determine horizontal asymptotes using the degree comparison rule
Distinguish between holes and vertical asymptotes by checking for common factors
Recognize slant asymptotes when the numerator degree exceeds the denominator degree by one
Interpret asymptotic behavior in real-world contexts

Quick Check

For the function $f(x) = \dfrac{(x+3)(x-1)}{(x+3)(x+5)}$, what occurs at $x = -3$?

Instructor Notes

Teaching Notes

Step 6 is a designed-failure step. Most students will predict that $x = 2$ is a vertical asymptote. The surprise when it turns out to be a hole is where the deepest learning happens. Do not hint at the answer beforehand.
The degree rule in step 5 is built inductively from three observations rather than stated as a rule. Students retain it better when they assemble it themselves.
The t-shirt hook connects to the Apply starter. You can reference it in class: "Remember the t-shirt problem? Now you know what that $12 ceiling is called."
The medication scenario in Apply Stretch is a good place to discuss what "approaching zero" means practically: the drug does not vanish instantly, but its concentration decays.

Common Student Errors

Every zero of the denominator is an asymptote. Step 6 directly addresses this. After the simulation, reinforce with additional examples that mix holes and asymptotes.
A graph can never cross a horizontal asymptote. Show examples like $\frac{x}{x^2+1}$, which crosses $y = 0$ at the origin.
Confusing the degree rule cases. Students sometimes apply the "ratio of leading coefficients" rule when degrees are different. The step 5 progression helps, but quiz questions should test all three cases.
Forgetting to check for common factors before identifying asymptotes. Teach the habit: factor first, cancel, then analyze.

Discussion Questions

Can a rational function have more than one horizontal asymptote? (No, but it can have different end behavior for $x \to +\infty$ vs $x \to -\infty$ if there is a slant asymptote.)
Is a hole "visible" on a calculator graph? Why or why not?
If you plug the hole's x-value into the simplified function, you get the hole's y-coordinate. Why does this work?
In the t-shirt problem, at what production volume does the average cost drop below $13? Below $12.50? Is there a practical limit to how close you can get to $12?

Exam Connection

Exam questions typically ask: "Find all vertical asymptotes, horizontal asymptotes, and holes." This simulation teaches the full procedure: factor, cancel, classify.
A common exam trap is a function like $\frac{x^2 - 9}{x^2 - 5x + 6}$ where one factor cancels (hole) and one does not (asymptote). Step 6 prepares students for this.
The degree rule for horizontal asymptotes appears on almost every exam covering rational functions. Step 5 ensures students can apply all three cases.