Knowledge Notes | Carey AP Precalculus Practice Lab

Cancel fully and compare remaining factor multiplicities before classifying the restriction.

What a strong response shows

Find every original denominator zero.
Compare complete factor multiplicities.
Use one-sided signs only when the question asks for them.

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What You Need to Know

Key idea

A vertical asymptote is a line $x=c$ where the rational function’s output has unbounded magnitude as $x$ approaches $c$ from at least one side.

Fast decision rule

Factor first. A denominator zero that remains after cancellation creates a vertical asymptote.

Worked example

If the reduced denominator contains $x-4$ , the graph has a vertical asymptote at $x=4$ .

Common mistake

Calling a cancelled denominator zero an asymptote.

Mini check

What must remain after factoring for an asymptote to exist?

Learning Objectives

Key idea

1.9.A determines vertical asymptotes of rational-function graphs.

Fast decision rule

Find denominator roots, remove cancelled roots, and write each remaining line as $x=c$ .

Worked steps

$1$ . Factor numerator and denominator. $2$ . Cancel common factors but record holes. $3$ . Solve the remaining denominator. $4$ . Check one-sided signs if asked.

Common mistake

Reporting the y-value of a branch instead of the x-value of the asymptote.

Mini check

Why is the answer written $x=c$ rather than $y=c$ ?

Key Concepts

Key idea

The original denominator identifies excluded inputs. After full reduction, use the signs of both numerator and denominator on each side to determine whether the branch tends to positive or negative $\infty$ .

Fast decision rule

Classify the input first, then use signs or a graph to describe one-sided behavior.

Worked example

At an uncancelled denominator root, the graph can rise on one side and fall on the other, or point the same way on both sides.

Common mistake

Assuming every asymptote has opposite signs on its two sides.

Mini check

What determines the sign of each one-sided branch?

Relevant Representations

Key idea

Factored form, a sign chart, and a graph all locate and describe vertical asymptotes.

Fast decision rule

Use the equation for the x-value, a sign chart for side behavior, and the graph for confirmation.

Classify denominator roots
Factor status	Graph feature	Report
uncancelled denominator factor	vertical asymptote	$x=c$
cancelled factor	hole	missing point
numerator zero only	x-intercept	$y=0$
no real root	no real vertical asymptote	none

Common mistake

Reading a horizontal asymptote from a vertical branch.

Mini check

Which coordinate stays fixed along a vertical asymptote?

Essential Relationships or Formulas

Key idea

Vertical asymptotes come from uncancelled denominator zeros.

r(x)=\frac{p(x)}{q(x)},\qquad q(c)=0

(x-c)\text{ remains in }q(x)\Rightarrow x=c\text{ is a vertical asymptote}

\text{After full reduction, denominator nonzero at }c\text{ and original denominator zero at }c\Rightarrow\text{hole at }x=c

Fast decision rule

Use the original expression to find restrictions, then use the reduced denominator to classify them.

Worked example

If the reduced denominator is a product of two linear factors, each real root gives a candidate vertical asymptote.

Common mistake

Skipping cancellation before classifying denominator roots.

Mini check

How many vertical asymptotes can a denominator with two distinct uncancelled real roots have?

Worked Examples

Key idea

Solve vertical-asymptote questions with a factor and sign workflow.

Fast decision rule

Separate location from one-sided behavior.

Worked steps

$1$ . Factor. $2$ . Cancel common factors. $3$ . Solve remaining denominator. $4$ . Write $x=c$ . $5$ . If needed, test one input just left and right.

Worked example

A graph approaching a fixed vertical line while its output grows in magnitude shows an asymptote; a smooth missing point does not.

Common mistake

Using a finite open circle as evidence of infinite behavior.

Mini check

What visual difference separates a hole from an asymptote?

Common Mistakes

Key idea

The main traps are cancellation, coordinate form, and one-sided sign interpretation.

Fast decision rule

Write the factor status next to every denominator root.

Worked example

A denominator root can disappear algebraically yet still remain a domain restriction; that case is a hole, not a vertical asymptote.

Common mistake

Forgetting that a vertical asymptote is a line, not a point.

Mini check

Does your answer use an equation of a line?

AP Exam Tips

Key idea

Most vertical-asymptote answers can be found by factoring before graphing.

Fast decision rule

Factor, cancel, solve the remaining denominator, and only then inspect branch direction.

Worked steps

$1$ . Find all original denominator roots. $2$ . Fully reduce common factors. $3$ . At each excluded input, a nonzero reduced denominator gives a hole; a zero reduced denominator gives a vertical asymptote. $4$ . Use one-sided signs when requested.

Common mistake

Listing every denominator root without checking cancellation.

Mini check

Did you state both the asymptote and any hole separately?

Practice

Use the Questionbank items below to determine vertical asymptotes and one-sided behavior.

TRY IT IN A NEW SETTING

Can you transfer the method?

Classify the exclusions of $r(x)=(x-2)(x+1)/((x-2)(x-4))$ .

Compare your reasoning

The original domain excludes $2$ and $4$ . Full reduction gives $\frac{x+1}{x-4}$ . At $x=2$ the reduced denominator is nonzero, giving a hole at $\left(2,\,\frac{-3}{2}\right)$ . At $x=4$ a denominator zero remains, giving the vertical asymptote $x=4$ .

Use it in a real question

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

Practise this topic Check the whole unit

1.9 · Rational Functions and Vertical Asymptotes

Topic 1.9 · Rational Functions and Vertical Asymptotes

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?