A.10(A)·Algebra II·Rational FunctionsSupporting

Rational Function Asymptotes

Use quotients of polynomials to describe rational graphs, domains, and asymptotes.

Students locate rational-function domain restrictions and vertical asymptotes by factoring, checking cancellation, and distinguishing denominator zeros from numerator zeros.

Lesson plan and worked examples by TEKSForge · not a TEA-issued lesson. Videos by independent educators.

Archived course: TEKS code and readiness label from TEA's 2013 Algebra II key. Algebra II is not one of Texas's current required STAAR EOC assessments, and this lesson does not change the Biology and Algebra I Study Plans.

Practice the released item Ask the tutor All Algebra II released practice50 minutes

Learn the concept

Original denominator sets the domain

A rational function is a quotient of polynomials. Exclude every input that makes its original denominator zero, even if simplifying later removes a factor.

Classify excluded inputs

After canceling common factors, a denominator zero is a vertical asymptote. If the denominator no longer vanishes at an excluded input, the graph has a removable hole there.

Keep graph features separate

A remaining numerator zero gives an x-intercept if it belongs to the domain. If the numerator degree is smaller than the denominator degree, the horizontal asymptote is $y=0$ . Vertical asymptotes have equations $x=c$ .

Worked example

Find the domain restrictions, vertical asymptotes, and x-intercept of $f(x)=\frac{x-2}{(x+7)(x-4)}$ .

Set the denominator factors equal to zero: $x+7=0$ or $x-4=0$ .
Exclude $x=-7$ and $x=4$ from the domain.
No factors cancel, so both excluded inputs give vertical asymptotes.
The numerator is zero at $x=2$ , where the denominator is nonzero.

Domain: $x\ne-7,4$ . Vertical asymptotes: $x=-7$ and $x=4$ . The x-intercept is $(2,0)$ .

Watch the lesson

Texas public-school, Texas educator-created, and TEKS-aligned lessons, hosted by their creators. Each card names its source.

Texas public school video

Algebra 2 Identifying Parent Functions

UT IPSI · Kid2Kid / Texas Education Agency

Recognize rational parent-function features, including asymptotic behavior.

Watch on YouTube

Try it yourself

TEKSForge practice · unscored

1. Find all vertical asymptotes of $g(x)=\frac{x+3}{(x-8)(x+2)}$ .
Show answer and reasoning
$x=8$ and $x=-2$
Both denominator factors remain after checking for cancellation.
2. Classify the excluded inputs of $h(x)=\frac{x-5}{(x-5)(x+1)}$ .
Show answer and reasoning
A hole at $x=5$ and a vertical asymptote at $x=-1$ .
The simplified rule is $\frac1{x+1}$ , but both original restrictions remain. The factor $x-5$ cancels completely.

Released STAAR practice

Texas Education Agency · archived 2013 released EOC items · scored with TEA's answer key; step-by-step explanations written by TEKSForge because this release has no rationales. Original graphs, tables, and diagrams are shown where the released item includes one.

Archived released STAAR Algebra II EOC · 2013 · Item 42 · 1 point

Which of the following gives the equations for all the vertical asymptotes of the graph of $f(x) = \frac{x + 4}{(x + 8)(x - 5)}$ ?

$x = -4$
$x = -5$ and $x = 8$
$x = -2$ and $x = 5$
$x = -8$ and $x = 5$

Show STAAR answer and explanation

$x = -8$ and $x = 5$

Vertical asymptotes occur where the denominator is 0 and the numerator is not. $x+8=0$ gives $x=-8$ , and $x-5=0$ gives $x=5$ .

$x = -4$ : $x=-4$ makes the numerator 0, giving an x-intercept, not an asymptote.

$x = -5$ and $x = 8$ : The signs are reversed; set each factor equal to 0.

$x = -2$ and $x = 5$ : $x=-2$ does not make the denominator 0.

Ask the tutor about this question

Start scored STAAR practice

Learning objectives

By the end of the lesson, students can:

Find excluded inputs from the original denominator.
Distinguish vertical asymptotes from removable holes.
Identify numerator zeros and basic end behavior.

Instructional sequence

1
Engage
5 min
Compare dividing by a very small number with dividing by zero.
2
Direct instruction
12 min
Model denominator zeros and check whether corresponding factors cancel.
3
Guided practice
13 min
Classify excluded values as asymptotes or holes and locate x-intercepts.
4
Graph check
13 min
Use nearby inputs and a graph to confirm behavior around excluded values.
5
Closure
7 min
Students list every vertical asymptote in the exit ticket.

Misconceptions to address

““A numerator zero creates a vertical asymptote.””

Teacher move: A numerator zero with a nonzero denominator creates an x-intercept.

““Every original denominator zero is a vertical asymptote.””

Teacher move: A fully canceled denominator factor creates a hole instead. Keep the original domain restriction.

Materials

•Factored rational-function cards
•Asymptote organizer
•Graphing calculator

Teaching tips

Factor completely before classifying excluded values.
Write asymptotes as equations of lines, not just numbers.
Use graphing technology to confirm algebra, since a graph may hide a hole.

Exit ticket · 3–5 min

Find every vertical asymptote of $f(x)=\frac{x+1}{(x-3)(x+6)}$ .

Show expected answer

$x=3$ and $x=-6$ . Neither denominator factor cancels with the numerator.

Sources & alignment

TEA 2013 Algebra II released test TEA 2013 Algebra II answer key and standards Texas Gateway, home of the UT IPSI Kid2Kid lessons

Alignment follows the standard TEA assigned to each 2013 released item. Video playback depends on YouTube access; written instruction remains available here.

Learn the concept

Original denominator sets the domain

Classify excluded inputs

Keep graph features separate

Worked example

Watch the lesson

Algebra 2 Identifying Parent Functions

Try it yourself

1. Find all vertical asymptotes of g(x)=x+3(x−8)(x+2)g(x)=\frac{x+3}{(x-8)(x+2)}g(x)=(x−8)(x+2)x+3​.

2. Classify the excluded inputs of h(x)=x−5(x−5)(x+1)h(x)=\frac{x-5}{(x-5)(x+1)}h(x)=(x−5)(x+1)x−5​.

Released STAAR practice

Archived released STAAR Algebra II EOC · 2013 · Item 42 · 1 point

Learning objectives

Instructional sequence

Engage

Direct instruction

Guided practice

Graph check

Closure

Misconceptions to address

1. Find all vertical asymptotes of $g(x)=\frac{x+3}{(x-8)(x+2)}$ .

2. Classify the excluded inputs of $h(x)=\frac{x-5}{(x-5)(x+1)}$ .