Lesson 2 | Quadratics | 11th Grade Mathematics

Objective

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.

Common Core Standards

Core Standards

The core standards covered in this lesson

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.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.

Foundational Standards

The foundational standards covered in this lesson

A.APR.A.1

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
F.BF.B.3

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.
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.

Criteria for Success

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

Describe the features that you can “see” in standard form and intercept form.
Calculate the vertex of a function represented in standard form by using $${-b\over{2a}}$$.
Identify the vertex of the function by knowing that the parabola is symmetrical around the vertex and then using the x-coordinate of the vertex to find the y-coordinate of the vertex.
Identify the y-intercept of the function in intercept form by finding the constant value.
Describe the purpose of using graphing calculators in Algebra 2.
Use [Y=] and [GRAPH] to graph quadratic functions.
Use [WINDOW] to adjust the viewing window. Adjust the scaling factor.
Use [Zoom] "0. ZoomFit" as well as other “Zoom” options to show quadratic.
Describe how the [CALC] and [TRACE] functions can be used to find features of quadratic functions on the graphing calculator.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson is the first in which graphing calculators are introduced.
This lesson may need to be split into two so you can adequately introduce the graphing calculator.
While Desmos is an invaluable tool that we will also use from this point forward, graphing calculators are essential tools for students to understand how to use for AP Calculus and other senior-level math courses as well as some college-level math.
The TI Connect CE software and the USB cable will be invaluable tools to allow for you to develop problem sets using graphing calculator screens.
This lesson provides an opportunity to discuss how to show work when a graph is produced on technology or information is determined from technology. Ensure there is common understanding of what students need to record to show evidence of learning.
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

Graph the following quadratic equation in your graphing calculator.

$${y=2x^2-4x+9}$$

Guiding Questions

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Problem 2

Below is a quadratic equation written in standard form.

$${y=2x^2-4x+9}$$

And this is the graph of the quadratic equation on a graphing calculator.

What do you need to change on the graphing calculator so you can “see” the y-intercept?
How can you use both the [ZOOM] and [WINDOW] features on the calculator to see this?
What options within the [CALC] feature will allow you to find the y-intercept?

Guiding Questions

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Problem 3

Part A

Find the vertex of the quadratic equation using the graphing calculator.

$${y=2x^2-4x+9}$$

Alter the window so you can more clearly see the vertex.
Use the [CALC] feature to determine the minimum value.

Part B

The formula $${-b\over2a}$$ can be used to find the axis of symmetry of the quadratic function. Use this to verify the vertex you found in the graphing calculator.

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

Below is a quadratic equation in standard form and the associated graph and window settings.

Find the vertex and the y-intercept algebraically.
Describe how you would find the vertex and the y-intercept on the graphing calculator.
Describe how you could adjust the graph so that the vertex, roots, and y-intercept are shown.

Lesson 1

Lesson 3

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

Interpreting Functions

Foundational Standards

Arithmetic with Polynomials and Rational Expressions

Building Functions

Seeing Structure in Expressions

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Lesson Map

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