Quadratic Equations and Applications

Students continue their study of quadratic equations, learning new strategies to determine the vertex and roots of quadratic equations and applying these in various real-world contexts.

Unit Summary

In Unit 8, Quadratic Equations and Applications, students continue their study of quadratic equations from Unit 7. They learn the three common forms of a quadratic equation—standard form, intercept form, and vertex form—and understand how to use these forms efficiently based on the situation at hand. Students also learn new strategies to determine the vertex and the roots of a quadratic equation and then apply these strategies in various real-world contexts. 

In Topic A, students are introduced to the vertex form of a quadratic equation. They use their factoring skills from Unit 7 to determine the process of completing the square. Using the process of completing the square, students are able to derive the famous quadratic formula, enabling them to solve for the roots of any quadratic equation. Students investigate examples of quadratic equations with two, one, and no real roots, and make the connection of the number of real roots to the value of the discriminant. Throughout the lessons in this topic, students pay attention to the structure of the equations to determine which strategy and approach are the most efficient way to solve. 

In Topic B, students recall how replacing the function $${{f(x)}}$$ with functions such as $${f(x+k)}$$ or $${{f(x)}}+k$$  transforms the graph of $${{f(x)}}$$ in predictable ways. Students then write and analyze quadratic functions to represent different real-world applications involving projectile motion, profit and revenue models, and geometric area applications. Lastly, students investigate systems of equations where one of the equations is a quadratic equation.

Assessment

This assessment accompanies Unit 8 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Essential Understandings

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  • A quadratic function can be written in three different forms (standard form, intercept form, and vertex form), where each form reveals different features of the function such as roots, vertex, $${y-}$$intercept, etc. 
  • The quadratic formula, $${x = {-b \pm \sqrt{{b^2-4ac}} \over 2a}}$$, can be derived from the general form of a quadratic equation, $${y=ax^2+bx+c}$$, and is used to determine the roots of any quadratic equation. The value of the expression under the square root, $${b^2-4ac}$$, called the discriminant, reveals the number of real roots of the quadratic equation. When the discriminant is positive, there are two real roots; when it is equal to $$0$$, there is one real root; and when it is negative, there are no real roots. 
  • Quadratic functions can be used to model a variety of real-world contexts such as projectiles and falling bodies, profit and revenue models, and geometric areas. The key features of quadratic functions, interpreted in context, provide information on values such as break-even points, revenue-maximizing selling prices, peak heights, and maximum measurements. 

Vocabulary

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Vertex form Quadratic formula
Intercept form Discriminant
Standard form Projectile motion
Complete the square Revenue

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: Deriving the Quadratic Formula

Topic B: Transformations and Applications

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.A.1.B — Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

  • 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.

Creating Equations
  • A.CED.A.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.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.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.

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

  • A.REI.B.4.A — Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)² = q that has the same solutions. Derive the quadratic formula from this form.

  • 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.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.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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Creating Equations
  • A.CED.A.4

Expressions and Equations
  • 8.EE.A.2

  • 8.EE.C.8

Geometry
  • 8.G.A.2

  • 8.G.A.3

Interpreting Functions
  • F.IF.A.2

  • F.IF.B.5

Reasoning with Equations and Inequalities
  • A.REI.C.6

Seeing Structure in Expressions
  • A.SSE.A.1

  • A.SSE.A.2

  • A.SSE.B.3

  • A.SSE.B.3.A

Future Standards

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Arithmetic with Polynomials and Rational Expressions
  • A.APR.B.3

High School — Number and Quantity
  • N.CN.A.1

  • N.CN.C.7

Interpreting Functions
  • F.IF.C.7

  • F.IF.C.7.C

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.