Algebra 1
Unit 3: Quadratic Functions and Modeling
In preparation for work with quadratic relationships students explore distinctions between rational and irrational numbers. They consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to the way in which extending the whole numbers to the negative numbers allows x+1 = 0 to have a solution. Formal work with complex numbers comes in Algebra II. Students expand their experience with functions to include more specialized functions—absolute value, step, and those that are piecewisedefined.
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What will my child learn?
Students will:
Part I: Graphical Analysis and Modeling of Quadratic Functions
Perform arithmetic operations on polynomials.
A.APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Focus on polynomial expressions that are linear or quadratic in a positive integer power of x.
Build a function that models a relationship between two quantities.
F.BF.A.1
Write a quadratic function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations.
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases (SAT® Content  PAM.13).
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
Interpret functions that arise in applications in terms of a context.
F.IF.B.4
For a quadratic 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, and end behavior.
F.IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationships it describes.
Build new functions from existing functions.
F.BF.B.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.
Summarize, represent, and interpret data on quantitative variables.
S.ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a linear function by plotting and analyzing residuals.
Part II: Algebraic Analysis of Quadratic Functions
Interpret the structure of expressions.
A.SSE.A.1
Interpret quadratic expressions that represent a quantity in terms of its context (SAT® Content  PAM.12).
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.
A.SSE.A.2
Use the structure of an expression to identify ways to rewrite it.
Write expressions in equivalent forms to solve problems.
Instructional Note: It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes handinhand with understanding what different forms of a quadratic expression reveal.
A.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines (SAT® Content  PAM.05  PAM.10).
Analyze functions using different representations.
F.IF.C.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 zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Write expressions in equivalent forms to solve problems.
A.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
Solve equations and inequalities in one variable.
Instructional Note: Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.
A.REI.B.4
Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as for real numbers a and b.
Represent and solve equations graphically.
A.REI.D.11
Explain why the xcoordinates 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, and rational functions.*
Students will:
Part I: Graphical Analysis and Modeling of Quadratic Functions
Perform arithmetic operations on polynomials.
A.APR.A.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Focus on polynomial expressions that are linear or quadratic in a positive integer power of x.
 Background Info.
 Check for Understanding: Adding and Subtracting Polynomials  Multiplying Binomials 1
 Enrichment Task: Powers of 11
Build a function that models a relationship between two quantities.
F.BF.A.1
Write a quadratic function that describes a relationship between two quantities.
a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
b. Combine standard function types using arithmetic operations.
 Background Info.
 Check for Understanding: Modeling with Composite Functions  Modeling with Sequences
 Enrichment Tasks: The Canoe Trip, Variation 1  Compounding with a 100% Interest Rate
Analyze functions using different representations.
F.IF.C.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases (SAT® Content  PAM.13).
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
 Background Info.
 Check for Understanding: Converting between PointSlope and SlopeIntercept Form  Converting between SlopeIntercept and Standard Form
Interpret functions that arise in applications in terms of a context.
F.IF.B.4
For a quadratic 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, and end behavior.
 Background Info.: Graphing and Interpreting Quadratic Functions*
 Check for Understanding: Interpreting Features of Functions  Positive and Negative Parts of Functions
 Enrichment Tasks: Warming and Cooling  Influenza Epidemic
F.IF.B.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationships it describes.
 Background Info.
 Check for Understanding: Domain and range from Graph  Domain of a Function
 Enrichment Tasks: Oakland Coliseum  Average Cost
Build new functions from existing functions.
F.BF.B.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.
 Background Info.
 Check for Understanding: Even and Odd Functions  Shifting and Reflecting Functions
 Enrichment Tasks: Transforming the Graph of a Function  Building a Quadratic Function from f(x) = x2
Summarize, represent, and interpret data on quantitative variables.
S.ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a linear function by plotting and analyzing residuals.
 Background Info.
 Check for Understanding: Analyzing Residuals  Linear Models of Bivariate Data
 Enrichment Tasks: Used Subaru Foresters I  Coffee and Crime
Part II: Algebraic Analysis of Quadratic Functions
Interpret the structure of expressions.
A.SSE.A.1
Interpret quadratic expressions that represent a quantity in terms of its context (SAT® Content  PAM.12).
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.
 Check for Understanding: Interpreting the Structure of Expressions  Structure in Expressions 1
 Enrichment Tasks: The Bank Account  Mixing Candles
A.SSE.A.2
Use the structure of an expression to identify ways to rewrite it.
 Background Info.
 Check for Understanding: Equivalent Forms of Polynomial Expressions  Factoring Difference of Squares 1
 Enrichment Tasks: Equivalent Expressions  Cubic Identity
Write expressions in equivalent forms to solve problems.
Instructional Note: It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes handinhand with understanding what different forms of a quadratic expression reveal.
A.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines (SAT® Content  PAM.05  PAM.10).
 Background Info.
 Check for Understanding: Solving Quadratics by Completing the Square 1
 Enrichment Tasks: Profit of a Company, Assessment Variation  Increasing or Decreasing? Variation 2
Analyze functions using different representations.
F.IF.C.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 zeroes, extreme values, and symmetry of the graph, and interpret these in terms of a context.
 Check for Understanding: Key Features of Quadratic Functions
 Enrichment Tasks: Which Function?  Springboard Dive
Write expressions in equivalent forms to solve problems.
A.SSE.B.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
 Background Info.
 Check for Understanding: Factoring Quadratics 1
 Enrichment Tasks: Profit of a Company, Assessment Variation  Increasing or Decreasing? Variation 2
Solve equations and inequalities in one variable.
Instructional Note: Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II.
A.REI.B.4
Solve quadratic equations in one variable.
b. Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as for real numbers a and b.
 Background Info.
 Check for Understanding: Quadratic Formula with Complex Solutions  Solving Quadratics by Taking the Square Root
 Enrichment Tasks: Visualizing Completion of the Square  Braking Distance
Represent and solve equations graphically.
A.REI.D.11
Explain why the xcoordinates 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, and rational functions.*
 Background Info.
 Check for Understanding: Intersecting Functions  Systems of Nonlinear Equations
 Enrichment Tasks: Introduction to Polynomials  College Fund  Ideal Gas Law
What are some signs of student mastery?
Graphical Analysis and Modeling of Quadratic Functions

Tools & Technology
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