Transcript Precalculus and Advanced Topics Module 3

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
A Story of Functions
Grade 12 Precalculus Module 3
Rational and Exponential Functions
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NYS COMMON CORE MATHEMATICS CURRICULUM
Participant Poll
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Session Objectives
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Participants will understand complex solutions to
polynomial equations.
Participants will understand function composition and
function inverses.
Participants will enrich their knowledge and experience
in order to implement Module 3 with confidence and
success.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Agenda
Today
• Module 3 Overview
• Module 3 Topic A – Polynomial Functions and the Fund. Theorem of Algebra
Tomorrow
• Module 3, Topic B – Rational Functions and Composition of Functions
• Module 3, Topic C – Inverse Functions
• Lunch
• Module 4, Topic A – Trigonometric Functions
• Module 4, Topic B – Trigonometry and Triangles
• Module 4, Topic C – Inverse Trigonometric Functions
• Discussion for Implementation
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 3: Rational and Exponential
Functions
Module Overview
• 3 Topics
• 21 Lessons
• 25 days
• Topic A – Polynomial Functions and the Fundamental Theorem of Algebra
• Mid-Module Assessment (After Topic A)
• Topic B – Rational Functions and Composition of Functions
• Topic C – Inverse Functions
• End of Module Assessment (After Topic C)
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 3: Rational and Exponential
Functions
Standards Addressed in this presentation:
• (+) N-CN.C.9:
Know the FTA; show valid for quadratic polynomials
• (+) A-APR.C.5:
Know and apply the binomial theorem for (x + y)n
• (+) G-GPE.A.3:
Derive equations of ellipses and hyperbolas
• (+) F-IF.C.7d:
Graph rational functions
• (+) F-BF-A.1c:
Compose functions
• (+) F-BF.B.4b:
Verify inverses by composition
• (+) F-BF.B.4c:
Read values of the inverse from a graph or table
• (+) F-BF-B.4d:
Restrict the domain to produce an invertible function
Others in the module: N-CN.C.8, G-GMD.A.2, A-APR.D.7, F-IF.C.9, F-BF-B.5
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Module 3: Rational and Exponential
Functions
Topic A
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Solving polynomial equations, and allowing complex number solutions.
Square roots of complex numbers and roots of unity.
Expanding binomial expressions using Pascal’s triangle.
Understanding and graphing ellipses and hyperbolas.
Topic B
• Rational functions and their graphs: end behavior and asymptotes.
• Function composition.
Topic C
• Inverse functions.
• Restricting the domain of a function to find an inverse.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic A: Polynomial Functions and the
Fundamental Theorem of Algebra
Topic Overview
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9 lessons (1-9).
Review FTA and establish for quadratic functions.
Explore square roots of complex numbers.
Consider roots of unity.
Know and be able to use the binomial theorem.
Understand the meaning of curves in the complex plane.
See ellipses and hyperbolas as curves in the complex plane and in
the Cartesian plane.
• Compute volumes by Cavalieri’s principle.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: The Fundamental Theorem of
Algebra
Opening Exercise: How many solutions
are there to the equation x2 = 1?
Question: Can you prove it?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: The Fundamental Theorem of
Algebra
Fundamental Theorem of Algebra:
1. Every polynomial function of degree n ≥ 1 with real
or complex coefficients has at least one real or
complex zero.
2. Every polynomial of degree n ≥ 1 with real or
complex coefficients can be factored into n linear
terms with real or complex coefficients.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: The Fundamental Theorem of
Algebra
Implications of the FTA:
If we allow complex numbers to be used as solutions
and coefficients, then:
1. Every polynomial equation of degree 1 or higher
has a solution.
2. Any polynomial of degree 1 or higher will factor
completely into linear terms.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 1: The Fundamental Theorem of
Algebra
A moment of proof: Prove that the FTA
holds in the case when n = 2.
That is, prove the following:
1. For a function f(x) = ax2 + bx + c, there is
at least one value of z so that f(z) = 0.
2. Every quadratic expression ax2 + bx + c
factors into linear terms.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Does Every Complex Number
Have a Square Root?
Let’s first think about this geometrically.
Recall: The modulus of the complex number
z = x + iy is the distance from the point
(x,y) to the origin:
The argument of the complex number z = x + yi is the amount that the positive
x-axis is rotated to align with the ray from the origin to the point (x,y), between
0 and 2p.
We often denote the modulus of a complex number by r and the argument by
q. Knowing the modulus and the argument uniquely defines the complex
number z.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Does Every Complex Number
have a Square Root?
Question: What does squaring a
complex number do to its modulus
and its argument?
Question: How can we find the square
root of a given complex number?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Does Every Complex Number
have a Square Root?
Question: How many square
roots does a nonzero complex
number have?
Describe them!
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Does Every Complex Number
have a Square Root?
The square root of a complex number z = x + yi is
another number p + qi:
1. Then (p + qi)2 = x + yi, so p2 - q2 + 2pqi = x + iy.
2. Equating real and imaginary parts gives p2 - q2 = x
and 2pq = y.
3. Solve the resulting system of equations.
4. Thus, to find a square root p + qi of z = x + yi , we
solve the two equations p2 - q2 = x and 2pq = y.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 2: Does Every Complex Number
have a Square Root?
Work through Exercise 8a in Lesson 2,
using the algebraic process to find the
square roots of z = 5 + 12i.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 3: Roots of Unity
Question: How many square roots of 1
are there?
Question: How many cube roots of 1
are there?
Work through the Exploratory
Challenge in Lesson 3 to find out.
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 3: Roots of Unity
For positive integers n, the nth roots of unity are the
solutions to the equation zn = 1 in the complex
numbers.
Question: How many nth roots of unity do you expect
there to be? Explain how you know.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 3: Roots of Unity
Work through Exercises 1-4 in Lesson 3
to explore:
• the square roots of unity
• the fourth roots of unity
• the sixth roots of unity
• the fifth roots of unity
Describe the roots of unity in both
rectangular and polar form.
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 3: Roots of Unity
Plot the nth roots of unity in the
complex plane for n = 4 and n = 6.
Make a conjecture: Describe the
location of the nth roots of unity in
the complex plane.
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: The Binomial Theorem
We start Lesson 4 by asking students to verify directly
that 1 + i and 1 - i are both solutions to the
polynomial equation
The conclusion drawn: the process of evaluating this
polynomial at a complex value of z is extremely
tedious.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 4: The Binomial Theorem
Pascal’s Triangle
The configuration of numbers below is known as
Pascal’s triangle. This triangle is easy to generate,
and surprisingly helpful when expanding binomial
expressions.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 4: The Binomial Theorem
The oldest known image of
Pascal’s Triangle is from 13th
century China, where it is still
called Yang Hui’s triangle. It was
known in China, Persia and
India as early as the 11th
century, AD.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: The Binomial Theorem
The entries in Pascal’s triangle can be found
recursively, as we have seen, but can also be found
directly using binomial coefficients.
The binomial coefficients
are the entries in row n, position k of Pascal’s triangle,
where we start counting with 0.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: The Binomial Theorem
Back to the original problem – how can
we speed up the process of expanding
(u + v)n?
Work through Exercises 8-11 in Lesson 4.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 4: The Binomial Theorem
The Binomial Theorem:
For any expressions u and v,
That is, the coefficients of the expanded binomial (u + v)n are
exactly the numbers in Row n of Pascal’s triangle.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: The Binomial Theorem
Work through Exercises 1 and 2 in Lesson 5.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 5: The Binomial Theorem
Work through the Exit Ticket from Lesson 5.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Curves in the Complex Plane
Question: What is the graph of the equation
Question: What is the graph of the equation
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Curves in the Complex Plane
Work through Exercises 1 and 2 in Lesson 6.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Curves in the Complex Plane
The anatomy of an ellipse:
Vocabulary: Ellipse, vertices, major
axis, semi-major axis, minor axis,
semi-minor axis, vertices center.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Curves in the Complex Plane
Work through Example 2 in Lesson 6.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 6: Curves in the Complex Plane
In the coordinate plane, ellipses centered at the origin
can be represented by the equation
where |a| is half of the length of the horizontal axis and
|b| is half the length of the vertical axis.
The points on an ellipse centered at the origin can be
written in polar form as
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Curves from Geometry
Watch the video of the Whispering Gallery in the
National Statuary Hall in the U.S. Capitol.
http://www.youtube.com/watch?v=FX6rUU_7kk.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Curves from Geometry
The docent demonstrates one geometric property of an
ellipse:
Every ray emanating from one focus of the ellipse
reflects off the curve in such a way that it travels to the
other focus of the ellipse.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Curves from Geometry
Thus, we can describe an ellipse as a set of points P in
the plane so that the sum of the distances from P to two
fixed points is constant.
The general equation for an ellipse centered at (h,k) is
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 7: Curves from Geometry
Do the Exercise:
Points F and G are located at (0,3) and (0,-3). Let P(x, y)
be a point so that PF + PG = 8. Use this information to
show that the equation of the ellipse is
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Curves from Geometry
When a satellite moves in a closed orbit around a planet, it
follows an elliptical path. however, if the satellite is moving
fast enough, it will overcome the gravitational attraction of the
planet and break out of its closed orbit. The minimum velocity
required for a satellite to escape the closed orbit is called the
escape velocity. The velocity of the satellite determines the
shape of its orbit.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Curves from Geometry
What happens to the graph if we change the equation
to
?
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Curves from Geometry
The geometric property of hyperbolas & ellipses:
Given two points F and G in the plane,
• An ellipse is the set of all points P in the plane so that
the sum of the distances PF + PG is constant.
• A hyperbola is the set of all points in the plane so that
the difference of the distances PF – PG is constant.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 8: Curves from Geometry
Work through Exercises 1-4, in which you will derive an
equation of a hyperbola given the foci, and graph
the resulting equation.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 8: Curves from Geometry
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NYS COMMON CORE MATHEMATICS CURRICULUM
Summing Up Topic A
Let’s Review
• What major concepts and/or
themes did you notice in Topic A?
• How do you think those topics
and themes will apply as we
move into studying rational
expressions and graphing rational
functions?
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic B: Rational Functions and
Composition of Functions
Topic Overview
• 8 lessons (10 - 17).
• Rational expressions and rational functions.
• Graphing rational functions: End behavior and asymptotes.
• Transformations of rational functions.
• Function composition.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 14: Graphing Rational Functions
Do Exercises 1-10, working in pairs.
Summarize the key features of the
graph of a rational function.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 14: Graphing Rational Functions
Key features of the graph of a rational function:
• x-intercepts
• y-intercepts
• end behavior
• vertical asymptotes
• horizontal asymptotes
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Function Composition
Function composition is introduced via an example:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Function Composition
Work through Example 1, which presents the ideas of
function composition before formally introducing
composition notation.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Function Composition
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Function Composition
Work through Example 2, in which
students interpret standard
composition notation in order to
determine which compositions make
sense.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 16: Function Composition
Work through Problem 2 on the Exit Ticket
for Lesson 16.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Solving Problems by Function
Composition
• Student outcomes:
•
•
•
write equations that represent functional relationships
and use the equations to compose functions.
analyze the domains and ranges of functions and function
compositions represented by equations, and
solve problems by composing functions.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 17: Solving Problems by Function
Composition
Work through Example 2 in Lesson 17.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Summing Up Topic B
Let’s Review
What major concepts and/or themes
did you notice in Topic B?
How have the topics and themes of
Topic B prepared you (and students)
to understand the upcoming work in
Topic C on Inverse Functions?
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Topic C: Inverse Functions
Topic Overview
• 4 lessons (18-21).
• Inverse functions.
• Restricting the domain.
• Inverse of logarithmic and exponential functions.
• Logarithmic and exponential problem solving.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 18: Inverse Functions
Work through Exercises 1-10 in Lesson 18.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 18: Inverse Functions
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lessons 18: Inverse Functions
Work through Exercises 11-13 in Lesson 18.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 18: Inverse Functions
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A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 18: Inverse Functions
Work through Problems 3a-d in the Problem Set.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 19: Restricting the Domain
A function is said to be invertible if its
inverse is also a function.
• Look at Exercise 7 and determine
which of the functions are invertible.
• Continue to Exercise 8, which is the
first instance where students need
to restrict the domain of a function
to make it invertible.
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NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Lesson 21: Logarithmic and Exponential
Problem Solving
In this lesson, students use radiocarbon dating to
estimate the time of death for wooly mammoth
specimens found in Siberia.
Look through the Exploratory Challenge for this lesson.
These exercises require number sense and quantitative
reasoning in addition to knowledge of inverses of
exponential functions.
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NYS COMMON CORE MATHEMATICS CURRICULUM
Summing Up Topic C
Let’s Review
What major concepts and/or themes
did you notice in Topic C?
How do those topics and themes rely
on topics studied earlier in the
module?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Functions
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Key Themes in Module 3: Rational and
Exponential Functions
• Working in the complex plane: Solving polynomial
equations, graphing functions, converting equations
for ellipses and hyperbolas between complex and
real form.
• Understanding rational expressions and graphing
rational functions.
• Composing functions and finding inverse functions.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Functions
Biggest Takeaway
•
What is your biggest takeaway from this study of
Module 3?
•
Given your role, how can you support successful
implementation at your school?
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