Analyzing the Graphs of Functions

Learning Resource Type

Classroom Resource

Subject Area

Mathematics

Grade(s)

9, 10, 11, 12

Overview

Given a set of information on the key properties of a function, you can sketch the graph. Before we proceed, make an attempt to summarize what you think are key properties. Often, the key properties of a function are not all presented to you directly but must be determined from the information at hand.

This informational material will explain how to analyze graphs of functions and identify the graph's key features. The article includes many examples of graphs and functions related to this concept. Practice questions with a PDF answer key are provided.

Mathematics (2019) Grade(s): 09-12 - Mathematical Modeling

MA19.MM.15

Use regression with statistical graphing technology to determine an equation that best fits a set of bivariate data, including nonlinear patterns.

UP:MA19.MM.15

Vocabulary

  • Regression Equation
  • "Best Fit"
  • Bivariate Data
  • Linear Pattern
  • Non-linear pattern
  • Scatter Plot
  • Quantitative Variable
  • Extrema
  • Inflection

Knowledge

Students know:
  • how to plot points using graphing technology.
  • how to find a regression equation using graphing technology.
  • how to use a regression equation to make a prediction.

Skills

Students are able to:
  • plot points.
  • Distinguish between linear and nonlinear functions.
  • Use graphing technology.

Understanding

Students understand that:
  • Regression equations can be used to model data.
  • Graphing technology helps us find regression equations.
  • The regression equation can be used to make a prediction.
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.
Mathematics (2019) Grade(s): 09-12 - Precalculus

MA19.PRE.27

Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.

UP:MA19.PRE.27

Vocabulary

  • Explicit expression
  • Recursive process
  • Compose functions

Knowledge

Students know:
  • Techniques for expressing functional relationships (explicit expression, a recursive process, or steps for calculation) between two quantities.
  • Techniques to combine functions using arithmetic operations.
  • Techniques to compose functions using algebraic operations.
  • Notation for function composition.

Skills

Students are able to:
  • Accurately develop a model that shows the functional relationship between two quantities.
  • Accurately create a new function through arithmetic operations of other functions.
  • Accurately create a new function through composition of other functions.
  • When functions are combined to create a new function, present an argument to show how the function models the relationship between the quantities.

Understanding

Students understand that:
  • Relationships can be modeled by several methods (e.g., explicit expression or recursive process).
  • Arithmetic combinations and/or composition of functions may be used to improve the fit of a model.

CR Resource Type

Informational Material

Resource Provider

CK-12

License Type

Custom
https://www.ck12.org/c/calculus/analyzing-the-graphs-of-functions/lesson/Analyz…

Custom License Type (URL)

https://www.ck12info.org/terms-of-use/

Accessibility

Text Resources: Content is organized under headings and subheadings

Approved Date

Owner (Author)

habradley
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