Knowledge Notes | Carey AP Precalculus Practice Lab

Compare degrees and leading coefficients to determine rational-function tails.

What a strong response shows

Record original-domain exclusions when cancelling.
Compare reduced numerator and denominator degrees.
Use the quotient to describe a slant or polynomial trend.

Loading the interactive investigation…

What You Need to Know

Key idea

A rational function’s end behavior depends on the degrees and leading terms of its numerator and denominator.

Fast decision rule

Compare degrees before doing detailed graph work.

Worked example

A denominator with higher degree pulls the ratio toward zero; equal degrees produce a constant horizontal level.

Common mistake

Comparing only constant terms.

Mini check

What degree comparison should you make first?

Learning Objectives

Key idea

1.7.A describes the end behavior of rational functions.

Fast decision rule

Use degree comparison, then division when the numerator degree is larger.

Worked steps

$1$ . Compare degrees. $2$ . If equal, divide leading coefficients. $3$ . If the numerator is larger, divide polynomials. $4$ . State the trend as $x$ grows in either direction.

Common mistake

Calling a polynomial trend horizontal.

Mini check

When is polynomial division required for end behavior?

Key Concepts

Key idea

Equal degrees give a horizontal asymptote; a numerator exactly one degree larger can give a slant asymptote; larger gaps can give a higher-degree polynomial trend.

Fast decision rule

The quotient from division describes the long-run trend when the numerator degree exceeds the denominator degree.

Worked example

A degree-two numerator over a degree-one denominator can approach a linear trend after division.

Common mistake

Calling every non-horizontal trend a slant asymptote.

Mini check

What does “slant” tell you about the quotient’s degree?

Relevant Representations

Key idea

Degree information, quotient form, and a graph should tell the same end-behavior story.

Fast decision rule

Use the equation to predict, division to reveal, and the graph to confirm.

Degree comparison guide
Degrees	Long-run result	Typical graph cue
numerator $<$ denominator	output approaches $0$	horizontal axis level
numerator $=$ denominator	ratio of leading coefficients	horizontal level
numerator $=$ denominator + $1$	linear quotient	slanted trend
numerator much larger	polynomial quotient	higher-degree trend

Common mistake

Ignoring a common factor before comparing the actual rational function’s behavior.

Mini check

Why does removing the same common factor preserve the numerator–denominator degree difference and end behavior, while the original excluded inputs remain?

Essential Relationships or Formulas

Key idea

Long-run rational behavior is encoded by degree comparison and polynomial division.

\deg p<\deg q\Rightarrow y=0

\deg p=\deg q\Rightarrow y=\frac{a_n}{b_n}

p(x)=q(x)s(x)+r(x),\quad \deg r<\deg q

Fast decision rule

Simplify common factors first, then compare the remaining degrees.

Worked example

If division gives a quotient $x-2$ plus a proper fraction, the graph follows the line $y=x-2$ as $x$ becomes large in magnitude.

Common mistake

Using the ratio of leading coefficients when the degrees are not equal.

Mini check

Which formula identifies the horizontal asymptote when degrees match?

Worked Examples

Key idea

Use one degree comparison or one division to make a precise tail statement.

Fast decision rule

Write the quotient or asymptote before describing the curve.

Worked steps

$1$ . Factor common terms if present. $2$ . Compare degrees. $3$ . Divide if needed. $4$ . Name the trend. $5$ . Check both directions if the question asks for end behavior.

Worked example

A rational curve can approach the same horizontal level from above on one side and below on the other; the asymptote is the level, not the side of approach.

Common mistake

Confusing the side of approach with the asymptote itself.

Mini check

Can a graph cross a horizontal asymptote?

Common Mistakes

Key idea

The common traps are degree comparison, cancellation, quotient degree, and one-sided language.

Fast decision rule

Circle the reduced numerator and denominator degrees before choosing a graph.

Worked example

If a common factor cancels, preserve the excluded input but use the reduced degrees for end behavior.

Common mistake

Keeping a cancelled factor in the reduced numerator or denominator degree count. Uncancelled factors still count.

Mini check

Which original-domain restrictions remain after cancellation, and why does cancellation preserve end behavior?

AP Exam Tips

Key idea

Rational end behavior can often be solved without graphing every branch.

Fast decision rule

Compare degrees, use the quotient when needed, and match the long-run trend to the graph.

Worked steps

$1$ . Reduce common factors. $2$ . Compare degrees. $3$ . Find the horizontal or polynomial trend. $4$ . Match both tails. $5$ . Separate end behavior from vertical asymptotes.

Common mistake

Answering a vertical-asymptote question when the prompt asks about $x$ becoming large.

Mini check

Did you use the phrase “as $x$ approaches $\infty$ ” or “as $x$ approaches negative $\infty$ ”?

Practice

Use the Questionbank items below to analyze rational-function end behavior.

TRY IT IN A NEW SETTING

Can you transfer the method?

Divide $(x^2+1)/(x-1)$ and identify the long-run line.

Compare your reasoning

The function equals $x+1+\frac{2}{x-1}$ , for $x\ne 1$ . Since the proper fraction tends to zero at either end, the slant asymptote is $y=x+1$ .

Use it in a real question

Work independently, then compare the mathematical evidence in your answer.

Practise this topic Check the whole unit

1.7 · Rational Functions and End Behavior

Topic 1.7 · Rational Functions and End Behavior

What a strong response shows

What You Need to Know

Learning Objectives

Key Concepts

Relevant Representations

Essential Relationships or Formulas

Worked Examples

Common Mistakes

AP Exam Tips

Can you transfer the method?