Lesson 3 | Quadratics | 11th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

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).

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

Seeing Structure in Expressions

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

Criteria for Success

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

Given a quadratic equation written in standard form and non-standard form; describe the process to factor a quadratic equation.
Identify the common factoring pattern of difference of two squares. Describe that this factoring pattern indicates symmetry around the y-axis.
Identify common factoring patterns of perfect square trinomials. Describe how this factoring pattern indicates a double root that is also the vertex of the function.
Use the roots to find the vertex of the function through analysis of the line of symmetry of the quadratic function.
Check solutions to problems using a graphing calculator.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson will require students to organize notes so they are able to easily see and access relevant information while studying. Students should annotate the equation with the ways they can find each of the features of the quadratic equation.

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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

Below is a quadratic equation.

$${g(x)=4x^2+7x-15}$$

What are the roots, vertex, and y-intercept of this function?

Guiding Questions

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Problem 2

Which equation matches with which graph? How do you know?

$${f(x)=-4x^2+8x-16}$$

$${g(x)=-4x^2+16}$$

Graph A

Graph B

Graph C

Graph D

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

Find the roots by factoring each of the following quadratic equations.

Additionally, for each equation, identify:

If the function has symmetry around the y-axis
If the root is a double root and the same as the vertex
The y-intercept
The vertex

a. $${3x^2-27=0}$$

b. $${2x^2+7x-15=0}$$

c. $${4x^2-10x+25=0}$$

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 that require students to identify errors in factoring—with signs, values, not removing a greatest common factor, and/or not identifying the difference of two squares.
Include problems where students are given the graph and asked to write the equation for the quadratic equation in intercept, standard, and vertex forms. Verify that the intercept form is correct by showing the calculations to factor the standard form to the intercept form.
Include problems where there is a missing value in the intercept form and students need to find the missing value based on constraints.
Include problems where students need to write a quadratic equation based on constraints (double root, symmetry around the y-axis, etc.).

Open Middle Factoring Quadratics
Kuta Software Free Algebra 2 Worksheets Free Algebra 2 Worksheets — Quadratic Functions and Inequalities: Factoring Quadratic Expressions and Solving Quadratic Equations by Factoring
Open Middle Multiplying Binomials
Open Middle Factoring Quadratics with Undefined C
Open Middle Minimum Value of a Quadratic in Factored Form

Lesson 2

Lesson 4

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

Interpreting Functions

Seeing Structure in Expressions

Seeing Structure in Expressions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Target Task

Additional Practice

Lesson Map

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