Download Course: Algebra 1 Unit #5: Quadratic Functions

Algebra 1
Quadratic Functions
Course: Algebra 1
Unit #5: Quadratic Functions
Overarching Question:
What patterns of change are modeled by quadratic functions as seen in real-world situations,
and the tables, graphs, and function rules that represent these situations?
Previous Unit:
Exponential Functions
This Unit:
Next Unit:
Quadratic Functions
Questions to Focus Assessment and Instruction:
1. What are the similarities and differences between quadratic, exponential,
and linear functions?
2. How do changes in the values of the parameters in a quadratic function
change the behavior of the graph?
3. What is the relationship between the number of real roots and the graph of
a quadratic equation? Why does this relationship exist?
4. How can you translate among the vertex, standard, and factored forms of
quadratic equations? What is useful about each form?
5. What are the characteristics of quadratic change as shown in a graph,
function rule, table, or real-world situation?
This document is the property of MAISA.
Solving Quadratic
Equations
Intellectual Processes (Standards for
Mathematical Practice):
Make sense of problems and persevere in
solving them: Build new mathematical
knowledge of quadratic patterns of change
through problem solving
Look for and make use of structure: Use
representations to model and interpret
physical, social, and mathematical situations
involving quadratic patterns of change
Attend to precision: Students communicate
about quadratic change in explaining their
reasoning precisely to others.
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Algebra 1
Quadratic Functions
Key Concepts:
Quadratic formula
Completing the square
Roots
Zeroes
Patterns of change in quadratic functions:
common second differences
Multiple representations
2
Vertex Form y = a(x - h) + k,
Factored(Root) Form y = a(x - p)(x - q)
2
Standard(Polynomial) Form y= ax + bx + c
Transformations of quadratic graphs based
on changes of a, b, and c
Key features of quadratic graphs: vertex,
axis of symmetry, minimum, maximum, xintercept(s), y-intercept
Modeling quadratic functions (key
applications: projectile motion and
profit/loss business models)
Compare and contrast quadratic functions
with other function families
Unit Abstract
An important nonlinear function category is quadratics. Understanding characteristics of quadratic
functions and connections between various representations should be developed in this unit. In
the table form of a quadratic function, the change in the rate of change distinguishes it from a
linear relationship. In particular, looking at the second rates of change or differences is where a
constant value occurs. The symmetry of the function values can be found in the table. The
graphical form shows common characteristics of quadratic functions including maximum or
minimum values, symmetric shapes (parabolas), location of the y-intercept, and the ability to
determine roots of the function. Quadratic functions can be written in a variety of formats:
polynomial form f (x) = ax2 + bx + c, factored form f (x) = a (x - p) (x - q), and vertex form f (x) = a (x
- h) 2 + k. The impact of changing the parameters a, b, c, h, k, p, and q should be explored and
understood. Connections should be made between each explicit form and its graph and table.
Real-world situations that can be modeled by quadratic functions include projectile motion,
television dish antennas, revenue and profit models in business, and the shape of suspension
bridge cables.
Common Core State Standards
Seeing Structure in Expressions (A-SSE)___________________________________________
Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For
example, interpret P(1+r)n as the product of P and a factor not depending on P.
Creating Equations (A-CED)______________________________________________________
Create equations that describe numbers or relationships
1. Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential
functions.
2. Create equations in two or more variables to represent relationships between quantities; graph
equations on coordinate axes with labels and scales.
Interpreting Functions (F-IF)______________________________________________________
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Algebra 1
Quadratic Functions
Interpret functions that arise in applications in terms of the context
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 periodicity.
5. Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the functionh(n) gives the number of person-hours it
takes to assemble n engines in a factory, then the positive integers would be an appropriate
domain for the function.
Analyze functions using different representations
7. 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.
8. Write a function defined by an expression in different but equivalent forms to reveal and explain
different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeros,
extreme values, and symmetry of the graph, and interpret these in terms of a context.
9. Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one
quadratic function and an algebraic expression for another, say which has the larger maximum.
Building Functions (F-BF)________________________________________________________
Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific
values of k (both positive and negative); find the value of k given the graphs. Experiment with
cases and illustrate an explanation of the effects on the graph using technology. Include
recognizing even and odd functions from their graphs and algebraic expressions for them.
Instructional Resources
NCTM Illuminations (http://illuminations.nctm.org)
Building Connections: This lesson focuses on having students make connections among
different classes of polynomial functions by exploring the graphs of the functions. The
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Algebra 1
Quadratic Functions
questions in the activity sheets allow students to make connections between the xintercepts of the graph of a polynomial and the polynomial's factors. This activity is designed
for students who already have a strong understanding of linear functions, some knowledge
of quadratic functions, and what is meant by a polynomial function.
http://illuminations.nctm.org/LessonDetail.aspx?id=L282
Egg Launch Contest: Students will represent quadratic functions as a table, with a graph,
and with an equation. They will compare data and move between representations.
http://illuminations.nctm.org/LessonDetail.aspx?id=L738
Texas Instruments (http://education.ti.com/calculators/timath/)
Bridge on the River Quad (TI-Nspire): In the first part of this activity, students graph a
quadratic function that models the shape of a bridge trestle. They then solve the related
quadratic equation by completing the square, recording each step as they complete it. This
list of steps is then generalized to deduce the quadratic formula. In the second part of the
activity, students store the formula in their handhelds, compare its results with those of the
nSolve command, and use it to solve several other quadratic equations.
http://education.ti.com/calculators/timath/US/Activities/Detail?sa=5022&id=9531
Car Stopping Distances (TI-84): This activity uses the transformation graphing application
on the TI-84 calculator to discover the equation for the stopping distance of a car on dry
pavement. http://education.ti.com/calculators/downloads/US/Activities/Detail?id=4070
Geometer’s Sketchpad
This applet displays parent functions for a variety of functions. In particular, choose the
quadratic tab. Sliders are set up in sketchpad format that allow manipulation of the
parameters a, b, and c in the standard form of a quadratic equation y = ax 2 + bx + c
http://mathbits.com/MathBits/GSP/functions.htm
PBS Mathline (http://www.pbs.org/teachers/classroom/9-12/math/resources/)
Getting Out of Line: This lesson emphasizes patterns, discovery, and vocabulary by
focusing on the basic connections between the graphs, tables, and symbolic
representations for lines, parabolas, inverse models, and exponential models. Students
investigate patterns and various models using the graphing calculator. It is designed for
first-year high school students and would be an ideal lesson following a unit on lines.
http://www.pbs.org/teachers/connect/resources/4374/preview/
Toothpicks and Transformations: The lesson begins with a review of transformations of
quadratic functions, followed by an activity where students explore a mathematical pattern
that emerges as they build a geometric design with toothpicks. Finally, the class uses
graphing calculators to develop another model and to verify that all of the models—factored
form, vertex form, and general form—are equivalent.
http://www.pbs.org/teachers/mathline/lessonplans/pdf/hsmp/toothpicks.pdf
Seeing Math
This applet allows the exploration of quadratic functions, by changing one of the
parameters, and seeing the impact on the parent function. It allows you to duplicate and
display multiple parabolas and their equations using a color-coded format. There is also a
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Algebra 1
Quadratic Functions
section that allows for transformations of the parabolas. Three different forms of quadratic
functions can be explored:
Polynomial or Standard form: y = ax 2 + bx + c ; Vertex Form: y = a(x ! b)2 + c ;
Root or Factored Form: y = a(x ! b)(x ! c)
http://seeingmath.concord.org/Interactive_docs/QT_Activity.pdf
Mathforum
Math Tools is a project of The Math Forum @ Drexel, funded in part by the National Science
Foundation. The goal is to create a community digital library that supports the use and
development of software for mathematics education.
http://mathforum.org/mathtools/sitemap.html
eNLVM: National Library of Virtual Manipulatives (http://enlvm.usu.edu/)
Transformations of families of functions including quadratic, absolute value, cubic, and linear
http://enlvm.usu.edu/ma/nav/toc.jsp?sid=__shared&cid=emready@trfns&cf=activity
Lesson plans to go with the eNLVM lesson above.
http://enlvm.usu.edu/ma/classes/__shared/emready@trfns/info/lessonplan.html
NCTM Reasoning and Sense Making Task Bank (www.nctm.org/hsfocus)
Horseshoes in Flight: Students analyze the structure of algebraic expressions and a graph
to determine what information each expression readily contributes about the flight of a
horseshoe. This task is particularly relevant to students who are studying (or have studied)
various quadratic expressions (or functions). The task also illustrates a step in the
mathematical modeling process that involves interpreting mathematical results in a realworld context.
http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSMTask/Horseshoes.pdf
Other Computer Applets
Ten quadratic topics explored in a variety of ways: Graph Transformations, Adding Functions
– The Polynomial Form, Products of Linear Functions, Quadratic Growth, Equivalent
Quadratic Expressions, Comparison and Operation, Solving Equations, Graphic Design,
Motion at Changing Speed, Economic Decisions
http://www.cet.ac.il/math/function/english/square/index.htm
This applet uses vertex form of a quadratic function y = a(x ! h)2 + k to have students
manipulate values of a, h, k using sliders to fulfill specific criteria. There are 3 questions
provided, but you could add more.
http://lasi.lynchburg.edu/peterson_km/public/old/java_examples/GENQUAD.HTM
There are two activities on this page. One explores parabolas by examining each variable in
the vertex-form equation. The second explores quadratic relationships through two concrete
examples of quadratic data.
http://a4a.learnport.org/page/quadratic-functions
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Algebra 1
Quadratic Functions
Move sliders to adjust the coefficients in the quadratic equation, both the Standard Form
and the Vertex Form. The goal is to hit the target parabola.
http://home.earthlink.net/~fossmountdesign/Applets/QuadraticsApplet.html
http://home.earthlink.net/~fossmountdesign/Applets/VertexApplet.html
Assessments
Students write quadratic equations to match the graphs.
http://itech.pensacolastate.edu/falzone/worksheets/transform-parabolas-notes.pdf
Find an equation of each parabola from the graph and put in the form: y = a(x-h)2 +k
http://itech.pensacolastate.edu/falzone/worksheets/graphing_transformation_wksht2.pdf
Use knowledge of vertices, minimum and maximum, x-intercepts, y-intercept, translations,
reflections to make an accurate sketch.
http://itech.pensacolastate.edu/falzone/worksheets/graphing_transformation_wksht1.pdf
Resources on algebra topics, explanation and practice
http://itech.pensacolastate.edu/falzone/course/1105handouts.htm
Professional Resources
NCTM (www.nctm.org)
Focus in High School Mathematics: Reasoning and Sense Making: This publication
elevates reasoning and sense making to a primary focus of secondary mathematics
teaching. It shifts the teachers’ role from acting as the main source of information to
fostering students’ reasoning to make sense of the mathematics.
http://www.nctm.org/catalog/product.aspx?ID=13494
Focus in High School Mathematics: Reasoning and Sense Making in Algebra:
Reasoning about and making sense of algebra are essential to students’ future success.
This book examines the five key elements (meaningful use of symbols, mindful manipulation,
reasoned solving, connection algebra with geometry, and linking expressions and functions)
identified in Focus in High School Mathematics: Reasoning and Sense Making in more detail
and elaborates on the associated reasoning habits.
http://www.nctm.org/catalog/product.aspx?ID=13524
Articles from National Council of Teachers of Mathematics (www.nctm.org)
Articles available as free downloads to NCTM members, or for a fee to non-members
Bergthold, T. (2005). Curve Stitching: Linking Linear and Quadratic Functions. Mathematics
Teacher, 98(5), 348-353. Retrieved February 18, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MT2005-01-348
Eraslan, A. and Aspinwall, L. (2007). Connecting Research to Teaching: Quadratic Functions:
Students’ Graphic and Analytic Representations. Mathematics Teacher, 101(3), 223. Retrieved
February 18, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MT2007-10-233a
This document is the property of MAISA.
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Algebra 1
Quadratic Functions
Edwards, T.G. and Ozgun-Koca, A. (2009). Technology Tips: Creating a Mathematics “B”
Movie: The Effect of nb on the Graph of a Quadratic. Mathematics Teacher, 103(3), 214.
Retrieved February 18, 2011 from
http://www.nctm.org/eresources/article_summary.asp?from=B&uri=MT2009-10-214a
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