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Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes
STRAND 3: QUADRATIC FUNCTIONS
TOPIC 3.3: APPLICATIONS
Topic Notes
Mathematical focus
Participants put quadratic functions to use in maximization problems, learn how
to extend standard textbook max/min problems to bring out the rich mathematics,
and discover a real world use for complex numbers.
Topic overview
This topic includes 3 tasks:
Task 3.3.1: Maximizing Revenue and Profit
Task 3.3.2: Making Application Problems Richer
Task 3.3.3: “Shocking” Numbers
These tasks bring together the knowledge the participants have developed about
quadratic functions. In Task 3.3.1, participants analyze ways to find a quadratic
equation using matrices and finite differences. The big idea here is to examine
various methods for defining a quadratic function given a pattern of a set of data.
In Task 3.3.2, participants get a chance to develop their own maximization
problems. Task 3.3.3 provides an example of the use of complex numbers in
electrical engineering, addressing the age-old question, “Does anybody actually
use this stuff?”
TExES standards focus
TExES Standard I.002 Number concepts. The teacher understands the
complex number system and its structure, operations, algorithms, and
representations. The beginning teacher:
(A) Demonstrates how some problems that have no solution in the real
number system have solutions in the complex number system.
(E) Describes complex number operations (e.g., addition, multiplication,
roots) using symbolic and geometric representations.
TExES Standard II.004 Patterns and algebra. The teacher uses patterns to
model and solve problems and formulate conjectures. The beginning
teacher:
(A) Recognizes and extends patterns and relationships in data presented in
tables, sequences, or graphs.
TExES Standard II.005 Patterns and algebra. The teacher understands
attributes of functions, relations, and their graphs. The beginning teacher:
(B) Identifies the mathematical domain and range of functions and
relations and determines reasonable domains for given situations.
(C) Understands that a function represents a dependence of one quantity
on another and can be represented in a variety of ways (e.g., concrete
models, tables, graphs, diagrams, verbal descriptions, symbols).
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
2
Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes
TExES Standard II.006 Patterns and algebra. The teacher understands linear
and quadratic functions, analyzes their algebraic and graphical
properties, and uses them to model and solve problems. The beginning
teacher:
(G) Models and solves problems involving linear and quadratic equations
and inequalities using a variety of methods, including technology.
TExES Standard V.018 Mathematical processes and perspectives. The teacher
understands mathematical reasoning and problem solving. The beginning
teacher:
(E) Understands the problem-solving process (i.e., recognizing that a
mathematical problem can be solved in a variety of ways, selecting an
appropriate strategy, evaluating the reasonableness of a solution).
TExES Standard V.019 Mathematical processes and perspectives. The teacher
understands mathematical connections both within and outside of
mathematics and how to communicate mathematical ideas and concepts.
(A) Recognizes and uses multiple representations of a mathematical
concept (e.g., a point and its coordinates, the area of a circle as a quadratic
function of the radius, probability as the ratio of two areas, area of a plane
region as a definite integral).
(B) Understands how mathematics is used to model and solve problems in
other disciplines (e.g., art, music, science, social science, business).
(C) Translates mathematical ideas between verbal and symbolic forms.
(D) Communicates mathematical ideas using a variety of representations
(e.g., numeric, verbal, graphical, pictorial, symbolic, concrete).
(E) Understands the use of visual media, such as graphs, tables, diagrams,
and animations, to communicate mathematical information.
TEKS/TAKS focus
TEKS 2A.1 Foundations for functions. The student uses properties and
attributes of functions and applies functions to problem situations.
The student is expected to:
(B) collect and organize data, make and interpret scatter plots, fit the
graph of a function to the data, interpret the results, and proceed to
model, predict, and make decisions and critical judgments.
TEKS 2A.2 Foundations for functions. The student understands the importance
of the skills required to manipulate symbols in order to solve problems and
uses the necessary algebraic skills required to simplify algebraic
expressions and solve equations and inequalities in problem situations.
The student is expected to:
(A) use tools including factoring and properties of exponents to simplify
expressions and to transform and solve equations; and
(B) use complex numbers to describe the solutions of quadratic equations.
TEKS 2A.3 Foundations for functions. The student formulates systems of
equations and inequalities from problem situations, uses a variety of
methods to solve them, and analyzes the solutions in terms of the
situations. The student is expected to:
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes
(B) use algebraic methods, graphs, tables, or matrices, to solve systems of
equations or inequalities; and
(C) interpret and determine the reasonableness of solutions to systems of
equations or inequalities for given contexts.
TEKS 2A.6 Quadratic and square root functions. The student understands
that quadratic functions can be represented in different ways and translates
among their various representations. The student is expected to:
(A) determine the reasonable domain and range values of quadratic
functions, as well as interpret and determine the reasonableness of
solutions to quadratic equations and inequalities; and
(B) relate representations of quadratic functions, such as algebraic, tabular,
graphical, and verbal descriptions.
TEKS 2A.8 Quadratic and square root functions. The student formulates
equations and inequalities based on quadratic functions, uses a variety of
methods to solve them, and analyzes the solutions in terms of the situation.
The student is expected to:
(A) analyze situations involving quadratic functions and formulate
quadratic equations or inequalities to solve problems.
High School TAKS Objective 1: The student will describe functional
relationships in a variety of ways.
High School TAKS Objective 2: The student will demonstrate an understanding
of the properties and attributes of functions.
High School TAKS Objective 5: The student will demonstrate an understanding
of quadratic and other nonlinear functions.
Materials
Task 3.3.1:
Maximizing
Revenue and
Profit
Graphing calculator
Chart paper
X
Task 3.3.2:
Making
Application
Problems
Richer
X
X
Task 3.3.3:
“Shocking”
Numbers
X
Procedure
These three tasks stand alone. Task 3.3.1 provides a nice review of matrix
algebra. If there is time for only one task from this topic, then Task 3.3.2 is the
most crucial, as it gives participants experience solving and enriching
maximization problems.
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
4
Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes
Summary
The big ideas here are to examine various methods for defining a quadratic
function given a pattern of a set of data. Participants use matrices, regression
equations and finite differences to find quadratic equations.
Assessment
What follows is an assessment that participants should be able to complete upon
finishing task 3.3.1.
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
5
Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes
Reflect and Apply
Consider the data below.
x
0
y
1.0
1
3.01
2
5.04
3
7.09
50
126.0
1. Use the first three points to find a quadratic function that fits the data. Check your
function using the 5th data point.
2. Find the formula for a linear function that passes through the second point and the
third point in the table.
3. Find the value of the linear functions at x = 3 and compare that value to the value of
the quadratic at x = 3 .
4. Compare the values of the linear and quadratic functions at x = 50 .
5. Use a graphing calculator to graph both functions on the same axes to estimate your
answers. Approximately what x-values would you have to use so that the resulting
function values for the quadratic function and the linear function would not differ
from each other by more than 0.5?
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.
6
Algebra II: Strand 3. Quadratic Functions; Topic 3. Applications; Topic Notes
Reflect and Apply Solutions:
2
1. y = 0.01x 2 + 2x + 1 , 0.01( 50 ) + 2 ( 50 ) + 1 = 126
2. y = 2.03x + 0.98 .
3. Linear function: x = 3, y = 7.07
Quadratic function: x = 3, y = 7.09
4. Linear function: x = 50, y = 102.48
Quadratic function: x = 50, y = 126
5. Participants may examine the table or graph the functions to
determine that this difference occurs for 0 ! x < 8.6 .
Task 3.3.2 is best suited to the teacher journal, whereas Tasks 3.3.1 and 3.3.3 would work
well with the student journal.
Teacher use only
Task 3.3.1
Task 3.3.2
Task 3.3.3
Modify for students
*
Ready for students
*
*
December 10, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University
of Texas at Austin for the Texas Higher Education Coordinating Board.