Use mathematical models to represent and understand quantitative relationships
Understand patterns, relations, and functions
Use graphs to analyze the nature of changes in quantitites
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.
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)
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.
Ask students to generalize how the applet can be used to find the roots of a quadratic equation.