Quadratic Functions and Solutions

Students investigate and understand the features that are unique to quadratic functions, and they learn to factor quadratic equations in order to reveal the roots of the equation.

Unit Summary

In Unit 7, Introduction to Quadratic Functions and Solutions, students take a closer look at quadratic functions. Because there is so much to cover on quadratic functions and equations, these concepts have been split over two units: Unit 7 and the last unit of the year, Unit 8. In Unit 7, students investigate and understand the features that are unique to quadratic functions, and they write quadratic equations into the equivalent intercept form in order to reveal the solutions of the equation. In Unit 8, students will learn about the vertex form and how to complete the square, along with digging into several real-world problems that involve quadratics.  

In Topic A, students analyze features of quadratic functions as they are seen in graphs, equations, and tables. They draw on their understandings of linear and exponential functions to compare how quadratic functions may be similar or different. 

In Topic B, students learn how to factor a quadratic equation in order to reveal the roots or solutions to the equation. They rewrite quadratic trinomials as the product of two linear binomials, and then using the zero product property, they determine the solutions when the function is equal to zero. Students also identify and compare solutions to quadratic functions that are represented as equations, tables, and graphs. Lastly, by determining the coordinates of the vertex of the parabola, students are able to sketch a reliable graph of the parabola using the $${x-}$$intercepts and the vertex as three defining points. 

In Topic C, students bring together the concepts and skills from the unit in order to interpret solutions to quadratic equations in context. They look at examples involving projectile motion, profit and cost analysis, and geometric applications. Students will spend more time with these applications in Unit 8.

Assessment

This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Essential Understandings

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  • Quadratic functions are represented as parabolas in the coordinate plane with a vertical line of symmetry that passes through the vertex. The roots or solutions of a quadratic function are the $$x$$-intercepts of the graph where $$f(x)=0$$, and can be determined algebraically using the equation and the Zero Product Property. 
  • Quadratic trinomials can sometimes be factored into the product of two linear binomials. Special factoring cases include a difference of two squares and perfect square trinomials. This factored form of a quadratic function, intercept form, is useful in revealing the zeros or solutions to a quadratic equation.

Vocabulary

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Quadratic functions Greatest common factor
Second difference Zero Product Property
Maximum/minimum Intercept form
Line of symmetry Linear binomial
Roots/solutions/$$x$$-intercepts Quadratic trinomial
Parabola Difference of two squares
Vertex Perfect square trinomial

Unit Materials, Representations and Tools

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  • Graphing technology

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 

Lesson Map

Topic A: Features of Quadratic Functions

Topic B: Factoring and Solutions of Quadratic Equations

Topic C: Interpreting Solutions of Quadratic Functions in Context

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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Arithmetic with Polynomials and Rational Expressions
  • 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.

  • A.APR.B.3 — Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Creating Equations
  • A.CED.A.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.

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.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(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. 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 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 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 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

  • 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.

Linear, Quadratic, and Exponential Models
  • F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

  • F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Reasoning with Equations and Inequalities
  • A.REI.B.4 — Solve quadratic equations in one variable.

  • 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.

Seeing Structure in Expressions
  • A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context 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.

  • A.SSE.A.1.A — Interpret parts of an expression, such as terms, factors, and coefficients.

  • 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 — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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.

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

Foundational Standards

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Expressions and Equations
  • 8.EE.A.1

  • 8.EE.A.2

Interpreting Functions
  • F.IF.A.2

Linear, Quadratic, and Exponential Models
  • F.LE.A.1

Future Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.C.4

  • A.APR.C.5

Building Functions
  • F.BF.B.3

Creating Equations
  • A.CED.A.2

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

  • A.REI.B.4.A

  • A.REI.B.4.B

  • A.REI.C.7

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

  • A.SSE.B.3.B

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.