Exploring Parabolas
by Suzanne Alejandre

## Teacher Lesson Plan

 NCTM Standards: Algebra:       Identify functions and contrast their properties from tables, graphs, or equations       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 quadratic equation that matches the graph of the corresponding parabola.       Identify the roots of the quadratic equation from viewing the graph.       Write the equation in factored form.       Make some generalizations based on the stated equations, given graphs and the two interactive graphs. Activity: Have 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. The following equations are displayed on the the green graph: y = x^2 - b * x + a (general) y = -x^2 + 5 * x - 6 y = x^2 - 14 * x + 48 y = x^2 + 14 * x + 48 y = x^2 + x - 2 -x^2 + 11 * x - 28 -x^2 - 8 * x - 15 The students' task is to match the correct quadratic equation to the matching parabola. As the students are trying to match the graph to the equation, the following questions may help prompt their thinking. Questions: Can you identify the roots of the equation based on the graph? Can you write the equation in factored form? What's the sum of the roots? What's the product of the roots? Can you write the equation in expanded form? Once students have matched the equations to the corresponding graphs, their next task is to find the coordinates on the blue graph that control the location of any of the given parabolas. Questions: What coordinates (a, b) can you find that move the black parabola to the position of one of the other parabolas? How many coordinates can you identify of the 6 graphs? Open the second 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. Questions: What coordinates (a, b) can you find that move the dark grey parabola to the position of one of the other parabolas? How many coordinates can you identify of the 6 graphs? Assessment: Have students generalize what they have observed as they have worked with the equations, the factored form, the roots, the graphs, and the coordinates (a, b).

 Math Forum Resources [T2T]  [Dr. Math] [Math Forum Resources] Selections from Teacher2Teacher Discussions: Parabolas Real application of parabolas. [Math Forum Resources] Selections from Ask Dr. Math Archives: Applications of Parabolas How are parabolas used in real life? Describing the Graph of an Equation Which of the following is true of the graph of the equation: y = 2x^2 - 5x + 3? Equation for an Arch I am trying to draw an arch that will go in the ceiling of a building. The arch will be at a maximum height of 28 inches... Graphing Parabolas How do you know how to graph a parabola from looking at its equation?