Lesson 5 | Quadratics | 11th Grade Mathematics

Objective

Solve quadratic equations written in vertex form and describe graphical features from vertex form.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

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.

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.

Criteria for Success

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

Describe the features that you can "see" easily in vertex form.
Convert vertex form to standard form to identify the y-intercept of a quadratic function.
Solve quadratic equations written in vertex form.
Identify the number and kind of roots of an equation in vertex form. Describe, with minimal calculation, which quadratic equations will result in non-real solutions.
Check solutions to problems using a graphing calculator.
Show that calculating $${\sqrt{-1}}$$ results in an "Error" on a graphing calculator. Describe why an error results.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

Allow students to have a graphing calculator to check their solutions and confirm their thinking of what the graph will look like.

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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

How would you find the roots of the following quadratic function without doing the following?

Converting to standard form and factoring
Using the quadratic formula
Graphing
Using a graphing calculator

$${f(x)=3(x+1)^2-12}$$

Guiding Questions

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Problem 2

John was solving the following quadratic equation:

$${f(x)={1\over2}(x+2)^2+6}$$

When he tried to find the decimal approximation for the roots, he got this message:

What went wrong?

Guiding Questions

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Target Task

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

Problem 1

Which of the following quadratic functions will result in non-real solutions?

$${{1\over2}(x-1)^2-5=0}$$

$${{1\over2}(x-1)^2+5=0}$$

$$-{{1\over2}(x-1)^2+5=0}$$

$$-{{1\over2}(x-1)^2-5=0}$$

Explain your resoning.

Problem 2

Solve the following quadratic function. Identify the number and kind of solutions.

$${g(x)={2\over3} \left({x-{1\over9}} \right)^2 -{8\over27}}$$

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 need to solve the vertex form and the roots are rational numbers, not integers.
Include problems where students are given an equation showing a difference of two squares. Ask students to find the solutions without factoring.
Include problems written in vertex form and ask students to identify all of the features of the quadratic function from the vertex form. Include a question similar to “How can you find the y-intercept without converting the equation to standard form?”
Include problems where students need to link the altering of an equation to form real solutions to the idea of transformations, and that real solutions result when a parabola crosses the x-axis.
Include problems such as: “Below is a screenshot of a quadratic function with the vertex noted but no roots shown in the window. Calculate the roots of the function using the vertex. The quadratic function is not vertically or horizontally dilated.”

Open Middle Quadratics with Defined Roots in Vertex Form

Lesson 4

Lesson 6

Lesson Map

Topic A: Features of Quadratic Functions

Identify features of quadratic functions from equations and use these features to graph quadratic functions. Derive the equation of a parabola.

F.IF.C.7.A F.IF.C.8.A G.GPE.A.2

Identify the y-intercept and vertex of a quadratic function written in standard form through inspection and finding the axis of symmetry. Graph quadratic equations on the graphing calculator.

F.IF.C.7.A F.IF.C.8.A

Write a quadratic equation in intercept form by factoring. Describe the features of a quadratic function written in intercept form.

A.SSE.A.2 A.SSE.B.3.A F.IF.C.8.A

Transform a quadratic function in vertex form. Describe the domain, range, and intervals where the function is increasing and decreasing.

F.BF.B.3 F.IF.B.4 F.IF.C.9

Solve quadratic equations written in vertex form and describe graphical features from vertex form.

A.REI.A.1 A.REI.B.4.B

Complete the square to convert an equation written in standard form to an equation written in vertex form.

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

Topic B: Imaginary Solutions and Operating with Complex Numbers

Identify solutions that are non-real from a graph and an equation using the discriminant. Define imaginary and complex numbers.

A.REI.B.4.B N.CN.A.1 N.CN.C.7

Add, subtract, and multiply complex numbers.

N.CN.A.1 N.CN.A.2

Topic C: Applications, Systems, and Inverse with Quadratics

Compare, analyze, and solve quadratic functions in projectile motion application problems.

F.BF.A.1 F.BF.B.3 F.IF.B.6 F.IF.C.8.A F.IF.C.9

Describe features of quadratic functions in the context of a word problem. Evaluate and use function notation to describe contextual situations.

F.BF.A.1 F.IF.B.6 F.IF.C.8.A F.IF.C.9

Identify solutions to a system of a quadratic function and a linear function graphically and algebraically.

A.REI.A.1 A.REI.C.7 A.REI.D.11

Quadratics

Objective

Common Core Standards

Core Standards

Reasoning with Equations and Inequalities

Reasoning with Equations and Inequalities

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Problem 1

Problem 2

Additional Practice

Lesson Map

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