Conquering Quadratics: Transforming to Vertex Form by Completing the Square

Algebra 2 Grades High School 3:34 Video
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Lesson Description

Master the art of converting quadratic equations into vertex form by completing the square. This lesson explores quadratic equations with leading coefficients of 1 and those not equal to 1. Discover how to identify the vertex and understand the parabola's characteristics.

Video Resource

Parabola Vertex Form for Quadratic Functions (by Completing the Square)

Mario's Math Tutoring

Duration: 3:34
Watch on YouTube

Key Concepts

  • Vertex Form of a Quadratic Equation
  • Completing the Square
  • Leading Coefficient Impact

Learning Objectives

  • Convert a quadratic equation from standard form to vertex form by completing the square.
  • Identify the vertex of a parabola given its vertex form equation.
  • Understand how the leading coefficient affects the shape and direction of a parabola.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the standard form of a quadratic equation (ax² + bx + c) and the vertex form (a(x - h)² + k), where (h, k) is the vertex. Briefly discuss why converting to vertex form is useful (e.g., easily identifying the vertex).
  • Completing the Square with a Leading Coefficient of 1 (15 mins)
    Referencing the video [0:12-2:33], explain the steps of completing the square when the leading coefficient is 1. Emphasize moving the constant term to the other side, taking half of the coefficient of the x term, squaring it, and adding it to both sides of the equation. Show how the perfect square trinomial factors into (x + b/2)².
  • Completing the Square with a Leading Coefficient Not Equal to 1 (20 mins)
    Using the video [2:33-4:08], demonstrate completing the square when the leading coefficient is not 1. Highlight the crucial step of factoring out the leading coefficient from the x² and x terms *before* completing the square. Stress the importance of distributing the factored-out coefficient when adding the squared term to both sides of the equation to maintain balance.
  • Examples and Practice (15 mins)
    Work through additional examples, varying the leading coefficient and the sign of the x term. Have students practice completing the square on their own with guided support.
  • Conclusion (5 mins)
    Summarize the steps for completing the square in both scenarios (leading coefficient of 1 and not equal to 1). Reiterate the importance of understanding vertex form and its connection to the parabola's vertex. Mention the relationship with graphing parabolas and other further topics.

Interactive Exercises

  • Group Problem Solving
    Divide students into small groups and assign each group a quadratic equation in standard form. Have them work together to convert it to vertex form by completing the square. Each group presents their solution to the class.
  • Error Analysis
    Present students with a worked-out example of completing the square that contains a common error (e.g., forgetting to distribute the leading coefficient). Ask them to identify and correct the mistake.

Discussion Questions

  • Why is it important to factor out the leading coefficient when completing the square?
  • How does the vertex form of a quadratic equation make it easier to graph the parabola?
  • Can you explain in your own words how to determine what number to add to both sides when completing the square?

Skills Developed

  • Algebraic Manipulation
  • Problem-Solving
  • Analytical Thinking

Multiple Choice Questions

Question 1:

What is the vertex form of a quadratic equation?

Correct Answer: a(x - h)² + k

Question 2:

In the vertex form a(x - h)² + k, what does (h, k) represent?

Correct Answer: The vertex

Question 3:

When completing the square, what do you do with the coefficient of the x term?

Correct Answer: Divide it by 2 and square it

Question 4:

If you add a value to one side of an equation, what must you do to the other side?

Correct Answer: Add the same value

Question 5:

What is the first step when completing the square if the leading coefficient is not 1?

Correct Answer: Factor out the leading coefficient from the x² and x terms

Question 6:

Given y = (x - 3)² + 5, what is the vertex?

Correct Answer: (3, 5)

Question 7:

The equation y = 2(x + 1)² - 4 represents a parabola that opens...

Correct Answer: Upwards

Question 8:

When completing the square, why is it important to factor out the leading coefficient before taking half of the 'x' term's coefficient?

Correct Answer: To ensure you are adding the correct value to both sides of the equation

Question 9:

What value of 'c' completes the square for x² + 6x + c?

Correct Answer: 9

Question 10:

After completing the square, the value outside of the parenthesis in the vertex form represent what point on the graph?

Correct Answer: The y value of the vertex

Fill in the Blank Questions

Question 1:

The process of rewriting a quadratic equation to easily identify the vertex is called completing the ________.

Correct Answer: square

Question 2:

In vertex form, y = a(x - h)² + k, the x-coordinate of the vertex is represented by _______.

Correct Answer: h

Question 3:

Before completing the square with a leading coefficient other than 1, you must _________ it out.

Correct Answer: factor

Question 4:

Taking one half of the 'x' coefficient and _________ it is a major step in completing the square.

Correct Answer: squaring

Question 5:

The standard form of a quadratic equation is written as y= ax^2 + bx + ________.

Correct Answer: c

Question 6:

Given the quadratic equation y=(x+4)^2 + 7, the vertex is (__________, 7).

Correct Answer: -4

Question 7:

The y value of the vertex of the quadratic y = -3(x-1)^2 + 5 is _________.

Correct Answer: 5

Question 8:

If a quadratic equation does not have a leading coefficient, that implies it's leading coefficient is _________.

Correct Answer: 1

Question 9:

When factoring out the leading coefficient, you only factor it from the x^2 and x term, and leave the ___________ alone.

Correct Answer: constant

Question 10:

The y value of the vertex will also be the equations maximum or ___________.

Correct Answer: minimum

Educational Standards

A2.A.1.1:Use mathematical models to represent quadratic relationships and solve using factoring, completing the square, the quadratic formula, and various methods (including graphing calculator or other appropriate technology). Find non-real roots when they exist. A2.F.1.3:Graph a quadratic function. Identify the domain, range, x- and y-intercepts, maximum or minimum value, axis of symmetry, and vertex using various methods and tools that may include a graphing calculator or appropriate technology. A2.A.2.2:Add, subtract, multiply, divide, and simplify polynomial expressions.

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