CramX Logo
RegisterLog in
Back to FlashcardsMathematics / AP Calculus AB: 8.5.3 Graphing Functions with Asymptotes

AP Calculus AB: 8.5.3 Graphing Functions with Asymptotes

Mathematics5 CardsCreated 8 months ago

This content guides how to graph rational functions by first identifying vertical and horizontal asymptotes through factoring, limits, and solving denominators. It also explains analyzing the function’s behavior around asymptotes using the first and second derivatives to determine increasing/decreasing intervals and concavity changes.

Report

Graphing Functions with Asymptotes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where
, or .
• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y = a, where , or .
• The behavior of a function can change from one side of a vertical asymptote to the other.

Rate to track your progress ✦

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/5

Key Terms

Term
Definition

Graphing Functions with Asymptotes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where th...

note

  • When sketching the graph of a rational function, you should first look for asymptotes.

  • Since this expression cannot be facto...

Which of the following curves is the graph of the equation

f(x) = 1/3x?

The graph of f(x)=13xf(x) = \frac{1}{3}x is a straight line passing through the origin with slope 13\frac{1}{3}.

Which of the following curves is the graph of the equation

f(x) = x/x−2

The graph of f(x)=xx−2f(x) = \frac{x}{x-2} has a vertical asymptote at x=2x=2 and a horizontal asymptote at y=1y=1, with the curve approaching but ...

Which of the following curves is the graph of the equation

f(x)= x/3−x?

The graph of f(x)=x3−x=−23xf(x) = \frac{x}{3} - x = -\frac{2}{3}x is a straight line through the origin with slope −23-\frac{2}{3}.

Related Flashcard Decks

Mathematics

Apex 2.2.3 Quiz: Graphing Functions

This deck covers key concepts from graphing functions, focusing on understanding graphical representations and their corresponding equations.

9 cards
View Deck
Mathematics

8 Basic Trigonometric Identities

This deck covers the fundamental trigonometric identities, including reciprocal, ratio, and Pythagorean identities.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.1 Antidifferentiation

This content provides an overview of antidifferentiation—the process of finding a function whose derivative is a given function. It introduces the concept of antiderivatives, outlines how to reverse common derivative rules, and includes simple examples to illustrate the idea.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.2 Antiderivatives of Powers

This content explains how to find antiderivatives of functions involving powers of 𝑥, emphasizing the concept of indefinite integrals and the constant of integration 𝐶. It highlights the idea that functions can have infinitely many antiderivatives and includes example problems demonstrating how to evaluate integrals involving powers and rational expressions.

9 cards
View Deck
Mathematics

AP Calculus AB: 9.2.2 Integrating Polynomials by Substitution

This content explains how to integrate polynomial expressions using integration by substitution, especially when dealing with composite functions. It outlines the step-by-step substitution process, emphasizes correct variable handling, and highlights the importance of rewriting the final answer in terms of the original variable along with including the constant of integration.

12 cards
View Deck
Mathematics

AP Calculus AB: 9.3.4 Choosing Effective Function Decompositions

This section focuses on how to strategically choose a substitution when performing integration by substitution. It explains that the best choice for 𝑢 is typically a part of the integrand whose derivative is also present, making the integral easier to evaluate. It also covers how to simplify trigonometric expressions using identities and emphasizes that effective decomposition is key to simplifying complex integrals.

6 cards
View Deck
Mathematics

AP Calculus AB: 9.3.1 Integrating Composite Trigonometric Functions by Substitution

This content covers the technique of integration by substitution applied to composite trigonometric functions. It explains how to identify the inner function for substitution, handle constant multiples, and ensure complete substitution. Worked examples demonstrate how substitution simplifies complex integrals, with emphasis on converting back to the original variable and verifying results through differentiation.

12 cards
View Deck
Mathematics

Conversion Factors and Problem Solving

This deck covers essential concepts related to rounding rules, significant figures, metric conversions, and basic geometry calculations.

18 cards
View Deck
Mathematics

Unit 1 Progress Check: MCQ Part A

This deck covers 18 questions and answers from a unit progress check, focusing on calculus concepts such as limits, derivatives, and function behavior.

18 cards
View Deck
AP Calculus AB: 8.5.3 Graphing Functions with Asymptotes
TermDefinition

Graphing Functions with Asymptotes

Identify vertical asymptotes for a rational function by factoring the numerator and denominator, canceling where possible, and determining where the resulting denominator is zero. A vertical asymptote to the graph of a function f is a line whose equation is x = a, where
, or .
• Identify horizontal asymptotes by taking the limit of the function as x approaches positive or negative infinity. A horizontal asymptote to the graph of a function f is a line whose equation is y = a, where , or .
• The behavior of a function can change from one side of a vertical asymptote to the other.

note

  • When sketching the graph of a rational function, you should first look for asymptotes.

  • Since this expression cannot be factored further and none of the factors cancel, set the denominator equal to 0. Solving for x indicates that the function has a vertical asymptote at x = 3.

  • Then take the limit of the function as x approaches . For the sake of this limit, you can ignore the constants. Canceling produces 1 for the limit. Thus the function has a horizontal asymptote at y = 1.

  • Next, you can find the first and second derivatives and determine the behavior of the function.

  • The first derivative of the function never equals 0. It is undefined at x = 3, but that is not a critical point because the function is not defined there. When you make your sign chart, mark x = 3 as the vertical asymptote.

  • The second derivative never equals 0 either. It is also
    undefined at x = 3, the vertical asymptote. Your sign chart should reflect this.

  • Although the concavity changes at x = 3, the function is not defined there, so there is no inflection point.

  • From the first derivative you can tell that the function is decreasing both to the left of the vertical asymptote and to its right. The second derivative indicates that the function is concave down on the left and concave up on the right.

Which of the following curves is the graph of the equation

f(x) = 1/3x?

The graph of f(x)=13xf(x) = \frac{1}{3}x is a straight line passing through the origin with slope 13\frac{1}{3}.

Which of the following curves is the graph of the equation

f(x) = x/x−2

The graph of f(x)=xx−2f(x) = \frac{x}{x-2} has a vertical asymptote at x=2x=2 and a horizontal asymptote at y=1y=1, with the curve approaching but never touching these lines.

Which of the following curves is the graph of the equation

f(x)= x/3−x?

The graph of f(x)=x3−x=−23xf(x) = \frac{x}{3} - x = -\frac{2}{3}x is a straight line through the origin with slope −23-\frac{2}{3}.