ALEX Classroom Resource

  

Precalculus and Advanced Topics Module 3, Topic B: Rational Functions and Composition of Functions

  Classroom Resource Information  

Title:

 Precalculus and Advanced Topics Module 3, Topic B: Rational Functions and Composition of Functions

URL:

 https://www.engageny.org/resource/precalculus-and-advanced-topics-module-3-topic-b-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 3, Topic B, students explore the composition of functions in-depth and notice that a composition of a polynomial function with the function f(x) = 1/x gives functions that can be written as ratios of polynomial functions. A study of rational expressions shows that these expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression. Students apply these operations to simplify rational expressions and go on to graphing rational functions, identifying zeros and asymptotes, and analyzing end behavior.

Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to Algebra I, Algebra II, and Algebra II with Trigonometry Alabama Courses of Study.

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra I
11 ) (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. [A-APR7]

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.A.HS.11- Add or subtract two polynomial expressions (limit to 2 terms each) with one variable.
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II with Trigonometry
30 ) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b]

b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [F-IF7c]

c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e]
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II with Trigonometry
33 ) Write a function that describes a relationship between two quantities.* [F-BF1]

a. Combine standard function types using arithmetic operations. [F-BF1b]

Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Mathematics
MA2015 (2016)
Grade: 9-12
Precalculus
19 ) (+) Compose functions. [F-BF1c]

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.
Mathematics
MA2015 (2016)
Grade: 9-12
Precalculus
18 ) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7]

a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b]

b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [F-IF7c]

c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [F-IF7d]

d. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e]
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
16. Compare and contrast relations and functions represented by equations, graphs, or tables that show related values; determine whether a relation is a function. Explain that a function f is a special kind of relation defined by the equation y = f(x).
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Can identify similarities and differences between relations and functions represented by equations, graphs or tables.
  • Determine if a relation is a function and explain that a function is a special kind of relation.
Teacher Vocabulary:
  • Function
  • Relation
Knowledge:
Students know:
  • How to represent relations and functions by equations, graphs or tables and can compare and contrast the different representations.
  • A function is a special kind of relation.
Skills:
Students are able to:
  • Compare and contrast relations and functions given different representations.
  • Identify which relations are functions and which are not.
Understanding:
Students understand that:
  • All functions are relations, but that some relations are not functions.
  • Equations, graphs, and tables are useful representations for comparing and contrasting relations and functions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.16.1: Define functions, relations (ordered pairs), input, output.
ALGI.16.2: Recall how to complete input/output tables.
ALGI.16.3: Recall how to read/interpret information from a table.
ALGI.16.4: Identify algebraic expressions.
ALGI.16.5 Recall how to name points from a graph (ordered pairs).
ALGI.16.6: Recall how to name points on a Cartesian plane using ordered pairs.

Prior Knowledge Skills:
  • Recall how to read a graph or table.
  • Define dependent variable, independent variable, ordered pairs, input, output, and coordinate plane.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.15 Use the vertical line test to determine if a given relation is a function.
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
17. Combine different types of standard functions to write, evaluate, and interpret functions in context. Limit to linear, quadratic, exponential, and absolute value functions.

a. Use arithmetic operations to combine different types of standard functions to write and evaluate functions.

Example: Given two functions, one representing flow rate of water and the other representing evaporation of that water, combine the two functions to determine the amount of water in a container at a given time.

b. Use function composition to combine different types of standard functions to write and evaluate functions.

Example: Given the following relationships, determine what the expression S(T(t)) represents.

Function Input Output
G Amount of studying: s Grade in course: G(s)
S Grade in course: g Amount of screen time: S(g)
T Amount of screen time: t Number of follers: T(t)
Unpacked Content
Evidence Of Student Attainment:
Students:
Given different types of standard functions
  • Use arithmetic operations to combine functions in context.
  • Use function composition to combine functions in context.
  • Write, evaluate, and interpret combined functions in context.
Teacher Vocabulary:
  • Function composition
Knowledge:
Students know:
  • Techniques to combine functions using arithmetic operations.
  • Techniques for combining functions using 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.
  • Present an argument to show how the function models the relationship between the quantities.
Understanding:
Students understand that:
  • Arithmetic combinations of functions may be used to improve the fit of a model.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.17.1: Define functions, relations (ordered pairs), input, output.
ALGI.17.2: Recall how to complete input/output tables.
ALGI.17.3: Recall how to read/interpret information from a table.
ALGI.17.4: Identify algebraic expressions.
ALGI.17.5: Recall how to name points from a graph (ordered pairs).
ALGI.17.6: Recall how to name points on a Cartesian plane using ordered pairs.

a.
ALGI.17.7: Identify, represent, and analyze two quantities that change in relationship to one another in real-world or mathematical situations.
ALGI.17.8: Set up an equation to represent the given situation, using correct mathematical operations and variables.

b.
ALGI.17.9: Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtration, and multiplication.

Prior Knowledge Skills:
  • Explain the distributive property.
  • Give examples of the properties of operations including distributive.
  • Combine like terms of a given expression.
  • Recognize the correct order to solve expressions with more than one operation.
  • Calculate a numerical expression (Ex. V=(4x4x4).
  • Choose the correct value to replace each variable in the algebraic expression (Substitution).
  • Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
21. Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Extend from linear to quadratic, exponential, absolute value, and general piecewise.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given relations or functions represented in various ways(algebraically, graphically, numerically in tables, or by verbal descriptions),
  • Use key features to compare the functions.
  • Explain and justify the similarities and differences of the functions.
Teacher Vocabulary:
  • Linear function
  • Exponential function
  • Quadratic function
  • Absolute value function
  • Linear Piecewise
Knowledge:
Students know:
  • Techniques to find key features of functions when presented in different ways.
  • Techniques to convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions).
Skills:
Students are able to:
  • Accurately determine which key features are most appropriate for comparing functions.
  • Manipulate functions algebraically to reveal key functions.
  • Convert a function to a different form (algebraically, graphically, numerically in tables, or by verbal descriptions) for the purpose of comparing it to another function.
Understanding:
Students understand that:
  • Functions can be written in different but equivalent ways (algebraically, graphically, numerically in tables, or by verbal descriptions).
  • Different representations of functions may aid in comparing key features of the functions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.21.1: Define function, function notation, (linear, polynomial, rational, absolute value, exponential, piecewise, and logarithmic) functions, and transitive property.
ALGI.21.2: Explain, using the transitive property, why the x-coordinates of the points of the graphs are solutions to the equations.
ALGI.21.3: Find solutions to the equations y = f(x) and y = g(x) using the graphing calculator.
ALGI.21.4: Solve equations for y.
ALGI.21.5: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations.

Prior Knowledge Skills:
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Graph a function given the slope-intercept form of an equation.
  • Demonstrate how to plot points on a coordinate plane using ordered pairs from a table.
  • Calculate a solution or solution set by combining like terms, isolating the variable, and/or using inverse operations.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.21 Given a function table, identify the missing number.

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
30. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph piecewise-defined functions, including step functions and absolute value functions.

c. Graph exponential functions, showing intercepts and end behavior.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a symbolic representation of a function (including linear, quadratic, absolute value, piecewise-defined functions, and exponential,
  • Produce an accurate graph (by hand in simple cases and by technology in more complicated cases) and justify that the graph is an alternate representation of the symbolic function.
  • Identify key features of the graph and connect these graphical features to the symbolic function, specifically for special functions:
      quadratic or linear (intercepts, maxima, and minima) and piecewise-defined functions, including step functions and absolute value functions (descriptive features such as the values that are in the range of the function and those that are not).
    1. Exponential (intercepts and end behavior).
Teacher Vocabulary:
  • x-intercept
  • y-intercept
  • Maximum
  • Minimum
  • End behavior
  • Linear function
  • Factorization
  • Quadratic function
  • Intercepts
  • Piece-wise function
  • Step function
  • Absolute value function
  • Exponential function
  • Domain
  • Range
  • Period
  • Midline
  • Amplitude
  • Zeros
Knowledge:
Students know:
  • Techniques for graphing.
  • Key features of graphs of functions.
Skills:
Students are able to:
  • Identify the type of function from the symbolic representation.
  • Manipulate expressions to reveal important features for identification in the function.
  • Accurately graph any relationship.
Understanding:
Students understand that:
  • Key features are different depending on the function.
  • Identifying key features of functions aid in graphing and interpreting the function.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.30.1: Define piecewise-defined functions and step functions.
ALGI.30.2: Graph functions expressed symbolically by hand in simple cases.
ALGI.30.3: Graph functions expressed symbolically using technology for a more complicated case.

a.
ALGI.30.4: Graph quadratic functions showing maxima and minima.
ALGI.30.5: Graph quadratic functions showing intercepts.
ALGI.30.6: Graph linear functions showing intercepts.

b.
ALGI.30.7: Define square root, cube root, and absolute value function.
ALGI.30.8: Graph piecewise-defined functions.
ALGI.30.9: Graph step functions.
ALGI.30.10: Graph cube root functions.
ALGI.30.11: Graph square root functions.
ALGI.30.12: Graph absolute value functions.

c.
ALGI.30.13 Identify exponential numbers as repeated multiplication.
ALGI.30.14 Rewrite exponential numbers as repeated multiplication.

Prior Knowledge Skills:
  • Demonstrate how to plot points on a coordinate plane using ordered pairs from a table.
  • Graph a function given the slope-intercept form of an equation.
  • Recognize the absolute value of a rational number is its' distance from 0 on a vertical and horizontal number line.
  • Define absolute value and rational numbers.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.12.30 Given the graph of a linear function, identify the intercepts, the maxima, and minima.

Tags: addition, asymptotes, division, functions, graph, multiplication, polynomials, ratio, rational expressions, rational numbers, subtraction
License Type: Custom Permission Type
See Terms: https://www.engageny.org/terms-of-use
For full descriptions of license types and a guide to usage, visit :
https://creativecommons.org/licenses
AccessibilityText Resources: Content is organized under headings and subheadings
Comments

There are eight lessons on this topic.

This resource is free for teachers to access and use. All resources required for the lessons are available to print from the site. 

Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to Algebra I, Algebra II, and Algebra II with Trigonometry Alabama Courses of Study.

  This resource provided by:  
Author: Hannah Bradley