Lesson 14 | Quadratic Equations and Applications | 9th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

Foundational Standards

The foundational standards covered in this lesson

8.EE.C.8

Expressions and Equations

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

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Understand the solution to a system is the point(s) of intersection of the graphs of the equations in the system.
Distinguish between the solutions to a quadratic equation (where the parabola crosses the $${x-}$$axis) and the solutions to a system that includes a quadratic equation (where the graphs of the two equations intersect).
Identify the solutions to a system that includes a quadratic and linear equation from a graph.
Determine the solutions to a system algebraically, where one of the equations in the system is quadratic and the other is linear.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Lessons 14 and 15 focus on systems of equations with quadratic and linear equations. In Lesson 14, all systems should have 2 solutions; Lesson 15 will address systems with 1 or 0 solutions.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

This diagram shows the graphs of $${{y=x^2}}$$ and $${ {y=2x}}$$. Fill in the labels to show which graph is which.

The graphs of $${{y=x^2}}$$ and $${y=2x}$$ cross each other at two points.

Write down the coordinates of these two points.
Show how you can use algebra to find the coordinates of the two points where the two graphs cross.

Guiding Questions

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References

Inside Mathematics Performance Assessment Tasks Grades 3-High School Graphs (2007) — Question #1 and #3

Graphs (2007) of the Performance Assessment Tasks created by the by the Mathematics Assessment Resource Service (MARS) of the Shell Centre for Mathematical Education, University of Nottingham, England are made available by Inside Mathematics under a license from Shell Centre Publications. Accessed July 12, 2018, 4:06 p.m..

Problem 2

The figure shows graphs of a linear and a quadratic function.

What are the coordinates of the point $$Q$$?
What are the coordinates of the point $$P$$?

Guiding Questions

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References

Illustrative Mathematics A Linear and Quadratic System

A Linear and Quadratic System, accessed on Aug. 18, 2017, 4:11 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Two functions are shown below.

$${f(x)=-x^2-2x+3}$$

$${{g(x)}=-3x-9}$$

a. What are the solutions to function $$f$$?

b. What are the solutions to the system of $$f(x)$$ and $${g(x)}$$? Give your answers as coordinate points.

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Include problems where students are given two equations, one of which is quadratic and the other is linear, and are asked to algebraically find the solutions to the system (only include examples where there are 2 solutions)
Include review problems of systems composed of a linear equation and an absolute value equation

Inside Mathematics Performance Assessment Tasks Grades 3-High School Graphs (2007) — (Complete the rest of the worksheet from Anchor Problem 1)
Inside Mathematics Performance Assessment Tasks Grades 3-High School Quadratic (2009)

Lesson 13

Lesson 15

Lesson Map

Topic A: Deriving the Quadratic Formula

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

Complete the square.

A.SSE.B.3.B

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

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

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

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

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

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

A.REI.B.4.A

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

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

F.IF.C.7.A

Topic B: Transformations and Applications

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

F.BF.B.3

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

F.BF.B.3

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

Write and analyze quadratic functions for geometric area applications.

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

Write and analyze quadratic functions for revenue applications.

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

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

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

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

Quadratic Equations and Applications

Objective

Common Core Standards

Core Standards

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Foundational Standards

Expressions and Equations

Reasoning with Equations and Inequalities

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem Set

Target Task

Additional Practice

Lesson Map

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