NCTM Standards:
Algebra:
      Use mathematical models to represent and understand quantitative relationships
      Understand patterns, relations, and functions
      Use graphs to analyze the nature of changes in quantitites

Objectives:
      Identify the roots of a quadratic equation from the graph and the factored equation.
      Understand connections between coefficients of the second and third terms of a quadratic equation and the roots of the equation.

Background:

Students learn to factor equations but often they don't have the conceptual understanding to accompany what they do mechanically with the numbers.

Given the equation of a parabola: x^2 - bx + a we start with an example with roots 2 and -3. Our equation is x^2 + x - 6.

Concepts to be emphasized during the activity include:

• 2 + (-3) = -1 or the coefficient of the second term, a
• (2)(-3) = -6 or the coefficient of the third term, b
• The sum of the two numbers is -1 and the product of the two numbers is -6.
• When a quadratic equation is graphed, it is a parabola
• The roots satisfy the equation so that y equals zero and therefore, most importantly, the roots of the equation can be read from the graph where the lines of the parabola cross the y axis.

The left grid is used to select a and b. Note what happens when (a, b) = (-1, -6)

Activity:

Students open the applet and get familiar with the controls.

Open the Java Applet

Note: It will open in a separate window. If you are displaying the page for students, arrange your browser windows so that the applet and the directions can be easily viewed. If students are working individually they should be encouraged to do this.

As students work through the activity they should:

• Realize that the black parabola is static but the location of (a, b) on the blue graph controls the location of the red parabola.
• Try the various points and possibly others as they try to control the red parabola.
• Recognize that the red (and black) parabola cross the x-axis at (2, 0) and also at (-3, 0).
• 2 and -3 are the satisfy the equation y = x^2 + x - 6.
• 2 and -3 are the roots of the quadratic equation y = x^2 + x - 6.
• See that when a = 0 and b = -4, then the parabola crosses the x-axis at (2, 0) and (-2, 0).
• Identify other values for a and b that determine the roots of the quadratic equation graphed on the green graph.

Assessment:

Ask students to generalize how the applet can be used to find the roots of a quadratic equation.

Math Forum Resources

[Problems of the Week]  [T2T]  [Dr. Math]

Selections from Math Forum Problems of the Week:

AlgPoW: What's My Angle? - posted September 18, 2000

Using concepts of vertical angles and supplementary angles, evaluate two quadratic expressions to find the measure of a given angle.

Selections from Teacher2Teacher Discussions:

Factoring

Which would be the best way to teach factoring?

Factoring and quadratics

Real life examples: area (tiling a floor), marketing (pricing tickets).

Parabolas

Real application of parabolas.

Selections from Ask Dr. Math Archives:

Analytic Geometry Formulas

Analytic geometry (a branch of geometry in which points are represented with respect to a coordinate system, such as cartesian coordinates) formulas for figures in one, two, and three dimensions: points, directions, lines, triangles, polygons, conic sections, general quadratic equations, spheres, etc.

Describing the Graph of an Equation

Which of the following is true of the graph of the equation: y = 2x^2 - 5x + 3?

Equations and Factoring

Solve. Identify all double roots: 2(r^2 + 1)=5r

Graphing Parabolas

How do you know how to graph a parabola from looking at its equation?

Learning to Factor

A sampling of answers from our archives.

What is a Quadratic Equation?

What is a quadratic equation? What is it used for? How can we use it to solve everyday problems?