Quadratics

Students build on their knowledge of the features, forms, and representations of quadratic functions, and extend their understanding from solutions in the real number system to the complex number system.

Unit Summary

In Unit 2, Quadratics, students review the features, forms, and representations of quadratic functions and extend their understanding from the solutions in the real number system to the complex number system. In this unit, students will also deepen their understanding of solutions of systems of quadratic equations and applications modeled with quadratic functions. 

Unit 2 begins with students identifying features of quadratic functions in multiple representations and converting between representations to reveal features, including transformations and symmetry, and connections between factoring and completing the square. Next in this unit, students will determine the number and kind of solutions using the discriminant, using graphical analysis, and by identifying non-real solutions found algebraically. Students will operate with complex numbers in Topic B as well, drawing connections to properties of operations in the real number system. In Topic C, students will extend the kinds of situations that can be modeled with quadratic functions from projectile motion, studied in Algebra 1, to geometric and profit function applications. Finally, students will review and extend their understanding of systems of quadratic equations; in the advanced course students will also explore systems with circles. If the advanced course is being taught, students will also investigate conjugates of complex numbers, write imaginary roots in intercept form, and restrict the domain of quadratics to render them invertible. 

As students continue their work in Algebra 2, analysis, transformations, systems, and inverse of functions and other equations will continue to be a theme. As more functions are added to the student repertoire, students will be expected to draw upon multiple understandings to solve problems.

Assessment

This assessment accompanies Unit 2 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

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Internalization of Standards via the Unit Assessment

  • Take unit assessment. Annotate for: 
    • Standards that each question aligns to
    • Purpose of each question: spiral, foundational, mastery, developing
    • Strategies and representations used in daily lessons
    • Relationship to Essential Understandings of unit 
    • Lesson(s) that assessment points to

Internalization of Trajectory of Unit

  • Read and annotate “Unit Summary."
  • Notice the progression of concepts through the unit using “Unit at a Glance.”
  • Do all target tasks. Annotate the target tasks for: 
    • Essential Understandings
    • Connection to assessment questions 

Unit-Specific Intellectual Prep

  • Give a factoring and completing-the-square pre-assessment to determine what knowledge and skills students have retained from Algebra 1. This will inform work done in Lessons 3 and 4. 
  • This unit is the first in which students are introduced to the graphing calculator. As you are reading through the lessons, think about how this new tool should be presented to students for proper and maximum use.

Essential Understandings

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  • Quadratic functions are represented graphically, algebraically in different forms, and contextually, to reveal features of the functions including maximum/minimum, intercepts, rates of change, and domain and range. 
  • Strategic use of algebraic structure, the properties of operations, and equality are used to identify solutions to quadratic equations and systems of equations using quadratic, absolute value, and circles. (adapted from SAT, Passport to Advanced Math) 
  • Knowledge of the complex number system is used to identify non-real roots, write quadratic equations with non-real roots in intercept form, and operate with complex numbers.

Unit Materials, Representations and Tools

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  • Desmos and the TI-84 will be used throughout this unit as graphing calculator tools.
  • The TI Connect CE software and the USB cable will be invaluable tools to allow for you to develop problem sets using graphing calculator screens.

Vocabulary

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roots, intercept, maximum, minimum, vertex standard, vertex, intercept (factored) form leading coefficient perfect square trinomial
double root difference of two squares equation of a circle: $${x^2+y^2=r^2}$$ invertible function
real and non-real solutions imaginary numbers complex numbers discriminant
complex roots tangent line inverse functions domain restriction

Graphing Calculator Skills

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Use [Y=] and [GRAPH] to graph
Use [WINDOW] to adjust the viewing window 
Use [Zoom] "0. ZoomFit" 
Use [Zoom] "6. ZoomStandard"
Use [CALC] and [TRACE] functions

Lesson Map

Topic A: Features of Quadratic Functions

Topic B: Imaginary Solutions and Operating with Complex Numbers

Topic C: Applications, Systems, and Inverse with Quadratics

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Building Functions
  • F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

High School — Number and Quantity
  • N.CN.A.1 — Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.

  • N.CN.A.2 — Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

  • N.CN.C.7 — Solve quadratic equations with real coefficients that have complex solutions.

Interpreting Functions
  • F.IF.B.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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.B.6 — Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

  • F.IF.C.7.A — Graph linear and quadratic functions and show intercepts, maxima, and minima.

  • F.IF.C.8.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.

  • F.IF.C.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.

Reasoning with Equations and Inequalities
  • A.REI.A.1 — Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

  • A.REI.B.4.B — Solve quadratic equations by inspection (e.g., for x² = 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 a ± bi for real numbers a and b.

  • A.REI.C.7 — Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x² + y² = 3.

  • A.REI.D.11 — 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. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Seeing Structure in Expressions
  • A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

  • A.SSE.B.3.A — Factor a quadratic expression to reveal the zeros of the function it defines.

  • A.SSE.B.3.B — Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Foundational Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.A.1

Building Functions
  • F.BF.B.3

  • F.BF.B.4.A

Creating Equations
  • A.CED.A.1

  • A.CED.A.2

  • A.CED.A.4

Expressing Geometric Properties with Equations
  • G.GPE.A.1

  • G.GPE.B.5

High School — Number and Quantity
  • N.RN.B.3

Reasoning with Equations and Inequalities
  • A.REI.B.4.B

  • A.REI.D.10

Seeing Structure in Expressions
  • A.SSE.B.3

Standards for Mathematical Practice

  • CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

  • CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

  • CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

  • CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

  • CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

  • CCSS.MATH.PRACTICE.MP6 — Attend to precision.

  • CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

  • CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.