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Project MathTalk

We (and Sasha and Keoni) just finished deriving the vertex form of a quadratic function

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Goals

Today we will develop geometric meaning for the standard form of a quadratic function (y = ax² + bx + c) and the quadratic formula
Our progression today

Reflect on the geometric information found in the vertex form
Re-express quadratics given in standard form into vertex form
Interpret the quadratic formula geometrically

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Group Work��Activity 1: �The Usefulness of the Vertex form of a Parabola

Project MathTalk

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Project MathTalk

Work together as a group to answer each question
Then prepare a presentation to the class that will involve all groups members presenting some part of the work

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Groups Share

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Rewriting an Equation Given in Standard Form into Vertex Form

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Project MathTalk

Suppose we want to find the same geometric information as in Activity 1:

Vertex
Line of symmetry
p
Focus
Directrix

That means we need to rewrite the equation into vertex form

Some of you may have learned a formula or procedure for this

Today, DON’T use formulas – use informal reasoning

First, we will do one together

Then you will work on three similar tasks for Activity 2

Parabola Given in Standard Form: y = x² + 6 x + 10

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Parabola Given in Standard Form: y = x² + 6 x + 10

Project MathTalk

Suppose we want to find the same geometric information as in Activity 1:

Vertex
Line of symmetry
p
Focus
Directrix

What is an equation that is close to the given equation but would be easy to rewrite in vertex form?

Rewrite your “close equation” in vertex form:

How does the original parabola compare to the parabola with the “close equation”

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Parabola Given in Standard Form: y = x² + 6 x + 10

Project MathTalk

y = x² + 6 x + 10 can be rewritten as

y = (x² + 6 x + 9) + 1

y = (x + 3)²+ 1

The equation is now in vertex form.

What is the vertex of the parabola?

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Group Work��Activity 2: �Rewriting the Equation for a Parabola from Standard to Vertex Form �

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Project MathTalk

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Groups Share

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Geometric Meaning for the Quadratic Formula

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Re-expressing the Equation from Standard to Vertex Form

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What is the vertex?

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Roots of a Quadratic Function

What does it mean to say something is a root of a quadratic function?

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Roots of a Quadratic Function

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Label the coordinate pairs that contain each root.

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Summarize in your own word: What does the quadratic formula mean?

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Homework 6

Covers material from today’s class
One of the questions asks you to rewrite y = x² - 4x + 5 in vertex form:

Sasha and Keoni solved the same task (and did so without a procedure or formula)
Homework 6 provides directions on how to watch this video, if you would like to

Project MathTalk

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The development of these videos were supported by the National Science Foundation through Awards DRL-1416789 and 1907782. The views expressed here are not necessarily those of the NSF

Acknowledgments

Dr. Joanne Lobato, PI, San Diego State University (SDSU)

Dr. John Gruver, Co-PI, Michigan Technological University (MTU)

Dr. Michael McKean, Technical Specialist, SDSU

Dr. Alicia Gonzales, Postdoctoral Scholar, MTU

Isabel White, Doctoral Student Research Assistant, SDSU & UCSD

Dr. Michael Foster, Postdoctoral Research Associate, Rochester Institute of Technology

Dr. C. David Walters, Creator of Teacher Support Materials, Weber State University