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EAST ALLEN COUNTY SCHOOLS
Bundle 5
Algebra II
Big Idea: Domain and Range Radical/Rational
Functions
Enduring Understandings Essential Questions
A composite function is a combination of two functions such that the output from the first function becomes the input for the second function.
The domain and range of a rational function can be found by hand, inspection, and via technology.
Graph rational functions by using asymptotes and intercepts and graph radical
functions by translations.
The inverse of a given function can be found graphically or algebraically.
Solve a radical equation by raising both sides of the equation to the same power.
Sove a rational equation by multiplying the equation by the least common
denominator.
Why do you rationalize a denominator?
How does a rational exponent correlate to its corresponding radical?
What is the domain of f(x)/g(x)?
How do you find the domain of a radical function without graphing?
How do you find the vertical and horizontal asymptotes?
What are the differences and similarities between direct and inverse variation?
CC/Learning Targets Core Vocabulary Links to Technology A.SSE.1b A.SSE.2 A.APR.6 A.REI.11 F.IF.4
F.IF.7 e F.LE.4 A2.6.1 A2.6.6 A2.10.1
inverse like radicals complex fraction point of discontinuity
EACS Rational Function Project
- iFactor (app) - MathRefFree (app) - AlgebraGenie (app) - KhanAcademy (app) - FreeGraphingCalculator (app)
Big Idea: Domain and Range Radical/Rational
Functions
Bundle Performance Task(s)
A utility company burns coal to generate electricity. The cost C (in dollars) of removing p amount (percent) of the smokestack pollutants is given by:80,000
(100 )
pC
p
Is it possible for the company to remove 100 percent of the pollutants? Discuss why or why not, and support your response by using algebraic analysis on the given model. What happens if the company does try to remove 100 percent of the pollutants? Will the company be successful at doing so, or will the attempt end in failure, that is, will it be to much expense for the company? Draw a diagram to show what the consequences of the last question would be. Lable the vertical asymptote(s) and analyze their impact on the company’s expense. You can use draw in MS Word to draw your diagram and submit. (There are also numerous interactive graphing resources on the Internet that can be used. Google it!) **If/when opening the pdf file, it may say that the file could conatin a virus etc. Ignore the warning and click “ok” to open the file** (See the attached PDF document to hand out to students.)
Algebra 2 Math Bundle 5
Quarter 3 Jan - Feb
Trigonometry
Algebra II Algebra II – Bundle 5
CC/Learning Targets Resource of Ideas Evidence of Learning A.SSE.1
(A2.2.4a,b) Interpret expressions that represent a
quantity in terms of its context.
-Prentice Hall Algebra II Lessons 7.1, 7.2, 7.3, 7.5, 7.6, 9.1, 9.6
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A.SSE.2 Use the structure of an expression to
identify ways to rewrite it. For example,
see x4 – y
4 as (x
2)2 – (y
2)2, thus
recognizing it as a difference of squares
that can be factored as (x2 – y
2)(x
2 + y
2).
-Prentice Hall Algebra II Lessons 9.3, 9.4, 9.5 - Pearson Algebra II iBook Lesson 8.4
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A.APR.6 (A2.5.2a)
Rewrite simple rational expressions in
different forms; write a(x)/b(x) in the
form q(x) + r(x)/b(x), where a(x), b(x),
q(x), and r(x) are polynomials with the
degree of r(x) less than the degree of
b(x), using inspection, long division, or,
for the more complicated examples, a
computer algebra system.
-Prentice Hall Algebra II Lesson 6.3 - Pearson Algebra II iBook Lesson 8.6
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A.APR.7
(A2.1.2a-c) 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.
-Prentice Hall Algebra II Lessons 8.1, 8.4, 8.5
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes
Trigonometry
Algebra II Algebra II – Bundle 5
-Bundle Task Assessments
A.REI.2 (A2.3.6a,b) (A2.6.4a) (A2.6.5a,b)
Solve simple rational and radical
equations in one variable, and give
examples showing how extraneous
solutions may arise.
-Prentice Hall Algebra II Lessons 8.5, 8.6 - Pearson Algebra II iBook Lesson 6.5
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A.REI.11 (A2.2.4c,d) (A2.3.7a,b) (A2.5.4a) (A2.5.5b) (A2.6.4a)
(A2.7.4a,b)
Explain why the x-coordinates of the
points where the graphs of the
equations y = f(x) and y = g(x) intersect
are the solutions of the equation f(x) =
g(x); find the solutions approximately,
e.g., using technology to graph the
functions, make tables of values, or find
successive approximations. Include
cases where f(x) and/or g(x) are linear,
polynomial, rational, absolute value,
exponential, and logarithmic functions.
-Prentice Hall Algebra II Lessons 7.7, 7.8, 9.2, 9.3, 9.6 - Pearson Algebra II iBook Lesson 8.6 First Concept Byte after Lesson 8.6 Second Concept Byte after Lesson 8.6 Concept Byte (after lesson 7.6)
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
F.IF.4
For a function that models a relationship
between two quantities, interpret key
features of graphs and tables in terms
of the quantities, and sketch graphs
showing key features given a verbal
description of the relationship. Key
features include: intercepts; intervals
where the function is increasing,
decreasing, positive, or negative;
relative maximums and minimums;
symmetries; end behavior; and
-Prentice Hall Algebra II Lessons 7.7, 7.8, 9.1, 9.2, 9.3 - MathGraph (app)
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
Trigonometry
Algebra II Algebra II – Bundle 5
periodicity.★
F.IF.7b (A2.1.1d-f) (A2.1.4a,b) (A2.1.6c) (A2.1.7a-d) (A2.2.1b,c)
Graph functions expressed symbolically
and show key features of the graph, by
hand in simple cases and using
technology for more complicated cases.
b. Graph square root, cube root, and
piecewise-defined functions, including
step functions and absolute value
functions.
-Prentice Hall Algebra II Lessons 7.4, 7.7, 7.8, 9.2, 9.3 - Pearson Algebra II iBook Lesson 6.8 Concept Byte after Lesson 8-1 Concept Byte after Lesson 8-3 - MathGraph (app)
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
F.BF.1 (A2.1.3a,b)
Write a function that describes a
relationship between two quantities.
b. Combine standard function types
using arithmetic operations. For
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.
-Prentice Hall Algebra II Lessons 7.4 - Pearson Algebra II iBook Lesson 6.6, 7.2
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
F.BF.4 Find inverse functions.
a. Solve an equation of the form f(x) = c
for a simple function f that has an
inverse and write an expression for the
inverse. For example, f(x) = 2x3 or f(x) =
(x + 1)/(x – 1) for x ≠ 1.
-Prentice Hall Algebra II Lessons 8.3, 8.5, 8.6 - Pearson Algebra II iBook Lesson 6.7, 7.3 Concept Byte (after lesson 6.7)
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A2.6.1 a. Simplify an expression with negative
exponents.
b. Simplify an expression with fractional
exponents.
-Prentice Hall Algebra II Lessons 7.4, 7.5
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources)
Trigonometry
Algebra II Algebra II – Bundle 5
-Test -Quizzes -Bundle Task Assessments
A2.6.2 a. Find a common denominator
between algebraic fractions.
b. Perform basic operation on algebraic
fractions.
c. Simplify algebraic fractions.
-Prentice Hall Algebra II Lessons 8.4, 8.5
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A2.6.6 a. Compare direct, inverse, and joint
variation.
b. Solve a problem involving direct
variation.
c. Solve a problem involving inverse
variation.
d. Solve a problem involving joint
variation.
e. Identify the type of variation present
in a problem.
-Prentice Hall Algebra II Lesson 9.1
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
A2.10.1 a. Use a variety of problem-solving
strategies, such as drawing a
diagram and writing an equation.
b. Use a variety of problem-solving
strategies, such as drawing a
diagram, guess-and-check, solving a
simpler problem, writing an equation,
and working backwards.
-Prentice Hall Algebra II Lessons 7.1, 7.2
-DMR -NearPod app -Varied Informal Assessments -Homework Packets (included in curriculum resources) -Test -Quizzes -Bundle Task Assessments
Trigonometry
Algebra II Algebra II – Bundle 5
Correlating CC/Learning Targets Teacher Notes
A.SSE.1b A.APR.1 A.CED.1
A.CED.2
A.CED.3 A.CED.4 F.FIF.5 F.IF.8 F.BF.3
- Pearson Algebra II iBook Complete Lesson 6.4 Lesson 8.2 (for CC standard A.APR.1) Lesson 8.6 (for CC standard A.CED.1 Lessons 8.1, 8.2 (For CC standard A.CED.2) Lessons 6.5, 8.1 (for CC standard A.CED.4)
Rational Functions Project Project for Rational Functions:
Goal: Analyze a Rational Functions and its graph. Objectives: • Recognize a rational function.
• Explain why the denominator of a rational function cannot be zero thus recognizing these values as the places where vertical
asymptotes occur and what they graphically look like.
• Student will explain why the values where vertical asymptotes appear are excluded from domain of the function and thus the graph does not touch or cross them.
• Student will demonstrate what the graph of the function does as it approaches the vertical asymptote from the left and right.
• Student will be able to graphically recognize what a horizontal asymptote is. Learning Objective: You will explain why the denominator of a rational function cannot be zero thus recognizing these values as the places where vertical asymptotes occur (which are disastrous things to have) and graphically what vertical asymptotes look like and mean.
Learning Activity: You will use a word problem (follows) with a grading rubric to explain the possible effects of dividing by zero. (This will be a real world application of a rational function.) Learning Activity: The following is an actual mathematical model used for Cost-Benefit analysis. The model is a rational function. Read the situation and analyze what the solution should be using the algebraic techniques we have studied.
1
Rational Functions Project Project for Rational Functions: Application of Rational Functions: If you want to know why it is important to understand Rational Functions, consider the following. This application is a Cost-Benefit Model. A utility company burns coal to generate electricity. The cost C (in dollars) of removing p amount (percent) of the smokestack pollutants is given by:
( ) 80,000(100 )
pC pp
=−
Is it possible for the company to remove 100 percent of the pollutants? Discuss why or why not, and support your response by using algebraic analysis on the given model. Remember to write in complete sentences. What happens if the company does try to remove 100 percent of the pollutants? Will the company be successful at doing so, or will the attempt end in failure, that is, will it be to much expense for the company? Remember to write in complete sentences. Make a graph to show what the consequences of the last question would be. Pick your scale carefully so that all the information you want to
discuss is visible on the graph. Remember to label the axes and show units and tick marks. Show the vertical asymptotes as dashed lines and lable them. Then discuss their impact on the company’s expense. You can use Geometer’s Sketchpad which is on most of the computers in the Go Center or draw in MS Word to draw your diagram and submit. (There are also numerous interactive graphing resources on the Internet that can be used. Google it!)
This project is slightly adapted from one written by a Professor Rust. I do not know who he or she is so I cannot give more complete credit than that.
Grading Rubric for Word Problem: Here is how I will assess your work:
2
Rational Functions Project Project for Rational Functions:
3
Name: ________________________ Teacher: Mr. Polley Date Submitted: ____________ Title of Work: ___________________
Criteria Points 4 3 2 1
Explanation A complete response with a
detailed explanation showing individual insight.
Response is a clear explanation, but no personal
in depth details. Explanation is unclear. Misses key points. ____
Use Of Visuals Clear diagram or sketch with details and labeling.
Diagram or sketch with no details or labeling.
Inappropriate or unclear diagram. No diagram or sketch. ____
Mechanics No math errors.
Complete sentences and properly constructed paragraphs
No major math errors, serious flaws in reasoning,
or major grammar and sentence structure problems
May be some serious math errors, flaws in reasoning, or grammar and sentence
structure mistakes
Major math errors, serious flaws in reasoning, major
grammar and sentence structure mistakes
____
Demonstrated Knowledge
Shows complete understanding of the questions, mathematical
ideas, and processes, gives individual insight to problem.
Shows understanding of the problem, ideas, and
processes, but no individual insight added only definitions given.
Response shows some understanding of the
problem.
Response shows a complete lack of
understanding for the problem.
____
Requirements Goes beyond the requirements of the problem, explains concepts
in detail enhancing answers with own insights and reasoning.
Meets the requirements of the problem, may explain
concepts by stating definitions, instead of
contributing own insights.
Hardly meets the requirements of the
problem.
Does not meet the requirements of the
problem. ____
Total----> ____ Teacher Comments:
Due 3/28 for 1st period and 3/29 for 7th period