Asymptotes and End Behavior

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

Most functions continue beyond the viewing window in our calculator or computer. People often draw an arrow next to a dotted line to indicate the pattern specifically. How can you recognize these asymptotes? 

 

This informational material will introduce students to the concept of vertical and horizontal asymptotes and explain how to identify asymptotes and the end behavior of functions from graphical representations. There is a corresponding video available. Practice questions with a PDF answer key are provided. 

Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.24

Compare and contrast families of functions and their representations algebraically, graphically, numerically, and verbally in terms of their key features. Note: Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; maximums and minimums; symmetries (including even and odd); end behavior; asymptotes; and periodicity. Families of functions include but are not limited to linear, quadratic, polynomial, exponential, logarithmic, absolute value, radical, rational, piecewise, trigonometric, and their inverses.

UP:MA19.PRE.24

Vocabulary

  • Function
  • Relative Maximum
  • Relative Minimum
  • Symmetry (Even/Odd)
  • End Behavior
  • Asymptotes
  • Intercepts
  • Increasing/Decreasing Intervals
  • Periodicity
  • Absolute Maximum
  • Absolute Minimum

Knowledge

Students know:
  • Properties of functions and make connections between different representations of the same function

Skills

Students are able to:
  • Compare properties of functions when represented in different ways (algebraically, graphically, numerically in tables or by verbal descriptions).

Understanding

Students understand that:
  • Each representation provides a unique perspective of the function.
  • Different representations are most appropriate for revealing certain key features of the function
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.26

Graph functions expressed symbolically and show key features of the graph, by hand and using technology. Use the equation of functions to identify key features in order to generate a graph.

UP:MA19.PRE.26

Vocabulary

  • Rational functions
  • Horizontal asymptote
  • Vertical asymptote
  • Slant asymptote
  • Amplitude
  • Period
  • Phase shift
  • Domain
  • Range
  • Frequency
  • Midline

Knowledge

Students know:
  • Techniques for graphing,
  • Key features of graphs of functions.

Skills

Students are able to:
  • Determine horizontal, vertical, and slant asymptotes of rational functions, and use these to sketch the graphs, identifydomains and ranges, and end behaviors.
  • Sketch the graphs, analyze, compare, and identify domains and ranges of the basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant.
  • Find the amplitude and period of a trigonometric function and use these characteristics to sketch its graph.
  • Identify and sketch translations of trigonometric graphs (vertical shifts and phase shifts).
  • Evaluate, graph and identify the domains and ranges of inverse trigonometric functions.

Understanding

Students understand that:
  • A rational function is the ratio of two polynomial functions.
  • Rational functions contain restrictions on their domains and/or ranges. Therefore, their graphs contain asymptotes, holes, and/or discontinuity.
  • The graphs of rational functions vary, yielding various patterns.
  • Using algebraic methods to manipulate and/or solve the equation of a rational function can help determine important properties such as its zeroes, intercepts, asymptotes, domain, range, types of discontinuity, and end behavior.
  • Key characteristics (rational and trigonometric) of functions can help you visualize the sketch of it's graph and can lead to more effective and efficient graphing methods.

CR Resource Type

Informational Material

Resource Provider

CK-12

License Type

Custom

Accessibility

Text Resources: Content is organized under headings and subheadings
Video resources: includes closed captioning or subtitles
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