CramX
RegisterLog in
Back to AI Flashcard MakerMathematics /AP Calculus AB: 8.5.1 Vertical Asymptotes

AP Calculus AB: 8.5.1 Vertical Asymptotes

Mathematics7 CardsCreated 3 months ago

This content explains how to identify vertical asymptotes in rational functions by factoring, canceling common factors, and setting the denominator equal to zero. It also distinguishes vertical asymptotes from holes, which occur when factors cancel in both numerator and denominator.

PrintEmbedImportReport

Vertical Asymptotes

  • Asymptotes are lines that the graph of a function approaches. A vertical asymptote to the graph of a function f is a line whose equation is x = a

  • Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero.

  • If a given point makes both the numerator and denominator of a function equal zero, then there might be a hole in the graph of the function at that point

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/7

Key Terms

Term
Definition

Vertical Asymptotes

  • Asymptotes are lines that the graph of a function approaches. A vertical asymptote to the graph of a function f is a line whose equation is...

note

  • The graphs of some rational functions have
    vertical asymptotes.

  • To determine vertical asymptotes you must follow three st...

Find the vertical asymptotes of the curve:

y = x−2 / x^2+4x+3.

x = −1 and x = −3

Find the vertical asymptotes.

y = x^2+2x / x^3−4x

x = 2

Find the vertical asymptotes of the curve:

y = x^2−x−6 / x^2+x−2.

x = 1

Find the vertical asymptotes.

y = x / x−2

x = 2

Log in to view all terms

Related Flashcard Decks

Mathematics

Essential SAT Math Formulas

This deck covers key formulas and concepts essential for the SAT Math section, including slope, distance, and vertex forms.

10 cards
View Deck
Mathematics

Algebra 1 Unit 1 Mastery Test

These properties of equality help maintain balance in an equation when performing the same operation on both sides. Inverse operations, like using addition to undo subtraction, are key tools in solving equations.

29 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 2

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

53 cards
View Deck
Mathematics

ATI TEAS Test Practice Mathematics Part 1

This flashcard set includes a variety of practical math questions covering percentages, conversions, averages, scale drawings, and basic algebra. It’s designed to strengthen problem-solving skills for real-life and standardized test situations.

55 cards
View Deck
Mathematics

Unit 2 Progress Check: MCQ Part A

To determine the point where the tangent line to the graph of the function 𝑓f has a slope of 2 for 𝑥 >0 x>0, we use the fact that the slope of the tangent line at any point is given by the derivative 𝑓′(𝑥)f′(x).

15 cards
View Deck
Mathematics

Customary Liquid Measurement Units

This set of flashcards outlines key relationships between customary units of liquid measurement, including pints, cups, quarts, and gallons. It provides conversion facts to help understand how these units relate to one another in the U.S. measurement system.

3 cards
View Deck

Study Tips

  • Press F to enter focus mode for distraction-free studying
  • Review cards regularly to improve retention
  • Try to recall the answer before flipping the card
  • Share this deck with friends to study together
AP Calculus AB: 8.5.1 Vertical Asymptotes
TermDefinition

Vertical Asymptotes

  • Asymptotes are lines that the graph of a function approaches. A vertical asymptote to the graph of a function f is a line whose equation is x = a

  • Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero.

  • If a given point makes both the numerator and denominator of a function equal zero, then there might be a hole in the graph of the function at that point

note

  • The graphs of some rational functions have
    vertical asymptotes.

  • To determine vertical asymptotes you must follow three steps. First, factor the numerator and denominator. Second, cancel any factors they have in common. Third, set the resulting denominator equal to zero and solve for x.

  • For the reciprocal function, no factoring or canceling is possible. When you set the denominator equal to zero, the result is x = 0. This is the equation of the vertical asymptote.

  • If a function has a vertical asymptote, its graph will get extremely close to the asymptote, but it will never cross it.

  • For the rational function on the far left, no factoring or canceling is possible.

  • When you set the denominator equal to zero, you get an equation that you can solve to produce x = 3. This is the equation of the vertical asymptote for the function.

  • The function on the near side is more complicated. It can be factored, and the numerator and denominator have a common factor. Canceling produces the same function as on the far left, except that it is not defined for x = 3. This produces a hole in the graph, not a vertical asymptote.

Find the vertical asymptotes of the curve:

y = x−2 / x^2+4x+3.

x = −1 and x = −3

Find the vertical asymptotes.

y = x^2+2x / x^3−4x

x = 2

Find the vertical asymptotes of the curve:

y = x^2−x−6 / x^2+x−2.

x = 1

Find the vertical asymptotes.

y = x / x−2

x = 2

Find the vertical asymptotes of f(x).

f(x) = x^2−x−2 / x^2+x−2

x = 1, x = −2