Quadratic Equations

A.Richter, Jun 14, 2016

Students will explore the graphs of quadratic equations.

Table of Contents

  1. Quadratic Equation Basics
    1. The Graph of a Quadratic Equation
    2. A Quadratic and Its Data Table
  2. Roots of Quadratic Equations
    1. The Roots of Two Quadratic Equations
    2. Finding the Roots of a Quadratic Equation
    3. Practice - Factoring to Find Roots
    4. Practice - Using the Quadratic Formula to Find Roots
  3. Maximum or Minimum of a Quadratic
    1. Maximum or Minimum of a Quadratic
    2. Finding the Maximum or Minimum
  4. Application of Quadratic Graphs
    1. Trajectory of an Object in the Air

1.

Quadratic Equation Basics

Students learn the basic elements of the graph of a quadratic equation.

  • 1. The Graph of a Quadratic Equation
  • 2. A Quadratic and Its Data Table

1.1.

The Graph of a Quadratic Equation

Move each slider to see how it changes the graph and equation given in both the standard and vertex forms.
– A.Richter

1.2.

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2.

Roots of Quadratic Equations

Students will learn to recognize where the real roots of a quadratic are located and when a quadratic has imaginary roots.

  • 1. The Roots of Two Quadratic Equations
  • 2. Finding the Roots of a Quadratic Equation
  • 3. Practice - Factoring to Find Roots
  • 4. Practice - Using the Quadratic Formula to Find Roots

2.1.

The Roots of Two Quadratic Equations

Drag the graph of each quadratic. Notice how the equation and the location of the roots change as the quadratic is dragged.
– A.Richter

2.2.

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2.3.

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2.4.

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3.

Maximum or Minimum of a Quadratic

Students will identify when a quadratic has a maximum or minimum.

  • 1. Maximum or Minimum of a Quadratic
  • 2. Finding the Maximum or Minimum

3.1.

Maximum or Minimum of a Quadratic

[math]y=ax^2+bx+c[/math] is the standard form of a quadratic equation. Move quadratic b. Quadratic b has a maximum point since its [math]ax^2[/math] term in negative. [br][br]Move quadratic c. Quadratic c has a minimum point since its [math]ax^2[/math] term in positive. [br][br]The dashed line going through the maximum or minimum point is the quadratic's axis of symmetry. The axis of symmetry divided the quadratic into equal halves.
– A.Richter

3.2.

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4.

Application of Quadratic Graphs

Students will learn about the most common application of the graph of a quadratic - trajectory of an object.

  • 1. Trajectory of an Object in the Air

4.1.

Trajectory of an Object in the Air

Move the sliders to change the quadratic.
A quadratic equation can represent the trajectory of an object thrown in the air. A trajectory is the path of an object while it is in the air, ending when it hits the ground or intended target.[br][br]Point A gives the starting height of the object the millisecond it is released. The [math]y[/math] value of point A is the starting height.[br][br]Point B is the vertex of the quadratic. Point B gives the maximum height of the object in the air. The [math]y[/math] value of point B is the maximum height.[br][br]Point C is one of the roots of the quadratic. Point C gives the maximum horizontal distance of the object. The [math]x[/math] value of point C is the total distance the object was thrown. When point C is on the [math]x[/math]-axis, it is considered ground level.[br][br]Move sliders a, b or c. [br]Slider a is the coefficient in the term [math]ax^2[/math].[br]Slider b is the coefficient in the term [math]bx[/math].[br]Slider c is the constant term (also the coefficient in the term [math]cx^0[/math]).
– A.Richter

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