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ALEX Classroom Resources  
   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 26 :
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.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
Subject: Mathematics (9 - 12)
Title: Analyzing the Graphs of Functions: Analyzing a Rational Function Interactive
URL: https://www.ck12.org/assessment/tools/geometry-tool/plix.html?eId=MAT.CAL.209.06&questionId=578565b88e0e080b52a6e3ec&artifactID=2777679&conceptCollectionHandle=calculus-::-analyzi&plix_redirect=1
Description:

This interactive will help students practice analyzing graphs of rational functions to identify key features. 

Analyzing functions helps one to derive key information that can be used to determine the equation of certain functions. In the interactive, students are given a rational function r(x) = 1 / x-h + k, where h and k are real numbers.

In this interactive, students will:


   View Standards     Standard(s): [MA2019] MOD-19 (9-12) 15 :
15. Use regression with statistical graphing technology to determine an equation that best fits a set of bivariate data, including nonlinear patterns.

Examples: global temperatures, stock market values, hours of daylight, animal population, carbon dating measurements, online streaming viewership

a. Create a scatter plot with a sufficient number of data points to predict a pattern.

b. Describe the overall relationship between two quantitative variables (increase, decrease, linearity, concavity, extrema, inflection) or pattern of change.

c. Make a prediction based upon patterns.
[MA2019] PRE-19 (9-12) 26 :
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.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
[MA2019] PRE-19 (9-12) 27 :
27. Compose functions. Extend to polynomial, trigonometric, radical, and rational functions.

Example: If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.
Subject: Mathematics (9 - 12)
Title: Analyzing the Graphs of Functions
URL: https://www.ck12.org/c/calculus/analyzing-the-graphs-of-functions/lesson/Analyzing-the-Graphs-of-Functions-CALC/?referrer=concept_details
Description:

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.


   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 6 :
6. Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.

[MA2019] PRE-19 (9-12) 26 :
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.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
Subject: Mathematics (9 - 12)
Title: Zeroes and Intercepts of Functions
URL: https://www.ck12.org/c/calculus/zeroes-and-intercepts-of-polynomials/lesson/Zeroes-and-Intercepts-of-Functions-PCALC/?referrer=concept_details
Description:

This informational material will explain how to identify the zeroes and intercepts of polynomial functions using graphs and equations. There are corresponding videos available. Practice questions with a PDF answer key are provided.  


   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 6 :
6. Analyze possible zeros for a polynomial function over the complex numbers by applying the Fundamental Theorem of Algebra, using a graph of the function, or factoring with algebraic identities.

[MA2019] PRE-19 (9-12) 26 :
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.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
Subject: Mathematics (9 - 12)
Title: Zeroes and Intercepts of Polynomials: Function Intercepts Interactive
URL: https://www.ck12.org/assessment/tools/geometry-tool/plix.html?eId=MAT.ALG.606&questionId=53da6ada5aa413702d860cd4&artifactID=1817921&conceptCollectionHandle=calculus-::-zeroes-and-intercepts-of-polynomials&collectionCreatorID=3&plix_redirect=1
Description:

This interactive will help students practice identifying zeroes and intercepts of polynomial functions using graphical representations. 

In this interactive, students will:


   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 24 :
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.
[MA2019] PRE-19 (9-12) 26 :
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.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
Subject: Mathematics (9 - 12)
Title: Asymptotes and End Behavior
URL: https://www.ck12.org/c/calculus/infinite-limit-type/lesson/Asymptotes-and-End-Behavior-PCALC/?referrer=concept_details
Description:
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. 

   View Standards     Standard(s): [MA2019] PRE-19 (9-12) 24 :
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.
[MA2019] PRE-19 (9-12) 26 :
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.

a. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

b. Graph trigonometric functions and their inverses, showing period, midline, amplitude, and phase shift.
Subject: Mathematics (9 - 12)
Title: Infinite Limit Type: Asymptotes and End Behavior Interactive
URL: https://www.ck12.org/assessment/tools/geometry-tool/plix.html?eId=MAT.CAL.105.03&questionId=57364d0c9616aa7d844c40bc&artifactID=2595735&conceptCollectionHandle=calculus-::-infinit&plix_redirect=1
Description:
In this interactive lesson, students will explore how asymptotes appear on the graphs of functions. Students will move the red lines to lie on the vertical and horizontal asymptotes of the function. 

ALEX Classroom Resources: 6

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