Algebra 1 | Quadratic Equations & Applications

Unit Summary

In Unit 8, 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.

Pacing: 17 instructional days (15 lessons, 1 flex day, 1 assessment day)

Fishtank Plus for Math

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Assessment

The following assessments accompany Unit 8.

Post-Unit

Use the resources below to assess student understanding of the unit content and action plan for future units.

Unit Prep

Intellectual Prep

Suggestions for how to prepare to teach this unit

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

Essential Understandings

The central mathematical concepts that students will come to understand in this unit

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

Terms and notation that students learn or use in the unit

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

Materials

The materials, representations, and tools teachers and students will need for this unit

Graphing technology

Lesson Map

Topic A: Deriving the Quadratic Formula

1

Describe features of the vertex form of a quadratic function and write quadratic equations in vertex form from graphs.

A.SSE.B.3 F.IF.B.4 F.IF.C.8

2

Complete the square.

A.SSE.B.3.B

3

Complete the square to identify the vertex and solve for the roots of a quadratic function.

A.REI.B.4.B A.SSE.B.3.B

4

Solve and interpret quadratic applications using the vertex form of the equation.

A.SSE.B.3.B F.IF.C.8.A

5

Convert and compare quadratic functions in standard form, vertex form, and intercept form.

F.IF.B.4 F.IF.C.9

6

Derive the quadratic formula. Use the quadratic formula to find the roots of a quadratic function.

A.REI.B.4.A

7

Determine the number of real roots of a quadratic function using the discriminant of the quadratic formula.

A.REI.B.4.B F.IF.C.7.A

8

Graph quadratic functions from all three forms of a quadratic equation.

F.IF.C.7.A

Topic B: Transformations and Applications

9

Describe transformations to quadratic functions. Write equations for transformed quadratic functions.

F.BF.B.3

10

Graph and describe transformations to quadratic functions in mathematical and real-world situations.

F.BF.B.3

11

Write and analyze quadratic functions for projectile motion and falling bodies applications.

A.CED.A.2 F.IF.C.8.A F.IF.C.9

12

Write and analyze quadratic functions for geometric area applications.

A.CED.A.2 F.IF.C.8.A

13

Write and analyze quadratic functions for revenue applications.

A.CED.A.2 F.BF.A.1.B F.IF.C.8.A

14

Solve and identify solutions to systems of quadratic and linear equations when two solutions are present.

A.REI.C.7 A.REI.D.11

15

Solve and identify solutions to systems of quadratic and linear equations when two, one, or no solutions are present.

A.REI.C.7 A.REI.D.11

Common Core Standards

Key

Major Cluster

Supporting Cluster

Additional Cluster

Core Standards

The content standards covered in this unit

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.

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.

Building Functions

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.

Building 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. 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.

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.

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.

Interpreting Functions

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.

Interpreting Functions

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.

Interpreting Functions

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.

Interpreting Functions

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.

Interpreting Functions

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.

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.

Reasoning with Equations and Inequalities

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.

Reasoning with Equations and Inequalities

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.

Reasoning with Equations and Inequalities

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.

Reasoning with Equations and Inequalities

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.

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.

Seeing Structure in Expressions

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

Standards covered in previous units or grades that are important background for the current unit

Creating Equations

A.CED.A.4

Creating Equations

A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

Expressions and Equations

8.EE.A.2

Expressions and Equations

8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.C.8

Expressions and Equations

8.EE.C.8 — Analyze and solve pairs of simultaneous linear equations.

Geometry

8.G.A.2

Geometry

8.G.A.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.A.3

Geometry

8.G.A.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Interpreting Functions

F.IF.A.2

Interpreting Functions

F.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F.IF.B.5

Interpreting Functions

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.

Reasoning with Equations and Inequalities

A.REI.C.6

Reasoning with Equations and Inequalities

A.REI.C.6 — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Seeing Structure in Expressions

A.SSE.A.1

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.2

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x<sup>4</sup> — y<sup>4</sup> 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

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.A

Seeing Structure in Expressions

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

Future Standards

Standards in future grades or units that connect to the content in this unit

Arithmetic with Polynomials and Rational Expressions

A.APR.B.3

Arithmetic with Polynomials and Rational Expressions

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.

High School — Number and Quantity

N.CN.A.1

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.C.7

High School — Number and Quantity

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

Interpreting Functions

F.IF.C.7

Interpreting Functions

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.C

Interpreting Functions

F.IF.C.7.C — Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

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.

Quadratic Equations and Applications

Unit Summary

Assessment

Post-Unit

Unit Prep

Intellectual Prep

Essential Understandings

Vocabulary

Materials

Lesson Map

Common Core Standards

Core Standards

Building Functions

Building Functions

Building Functions

Building Functions

Creating Equations

Creating Equations

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Interpreting Functions

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Foundational Standards

Creating Equations

Creating Equations

Expressions and Equations

Expressions and Equations

Expressions and Equations

Geometry

Geometry

Geometry

Interpreting Functions

Interpreting Functions

Interpreting Functions

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Seeing Structure in Expressions

Future Standards

Arithmetic with Polynomials and Rational Expressions

Arithmetic with Polynomials and Rational Expressions

High School — Number and Quantity

High School — Number and Quantity

High School — Number and Quantity

Interpreting Functions

Interpreting Functions

Interpreting Functions

Standards for Mathematical Practice

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy