Mr. Fizzix Physics Nov. 3, 2000   Joanne Pirog                                         Lesson Plan I Frameworks‑ Patterns, Relations and Algebra Analyze and graph Quadratic functional relationships   Pass out graph paper to the class. Ask them to investigate what the graph of y=lx^2 looks like by substituting lots of points for x. Ask them what shape their graph looks like? Tell them that this u‑shaped graph is called a parabola. Next, have the students graph y‑‑ ‑1x ^2. Ask them how this graph looks different than the first parabola they graphed? With several other examples have the students try to determine what effect a positive coefficient of the x^2 term vs. a negative coefficient of the x^2 term has on the graph of the parabola. A positive coefficient makes the curve look like Mwhile a negative coefficient makes the curve look like L                                                                                                                             An important lesson from this activity for the Trebuchet Project will be for them to realize that the graph of the path of a projectile launched by the Trebuchet will be that of an upside‑down u‑shaped parabola. They will soon realize that the graph of the parabola y = ‑2x^2 + x will be this upside‑down u‑shape.   Next explain to the class how these upside down parabola graphs have a maximum vertex point. Demonstrate with a diagram how the projectile will achieve a maximum height in the air before coming down again. Graph examples of parabolas such as y = ‑ I x^2+1 and y = ‑1 x^2 ‑2 and have the class find all the maximum vertices of the graphs.   Finally explain to the students that the actual path of a projectile launched by the Trebuchet is L ‑16T^2+ (v sin theta) T = 0. It is the ‑16 coefficient that produces the   L      path. T = 5 sec, theta is the angle of release of the arm and H=O when the projectile lands again. The best angle of release is supposedly an angle of 45 degrees because the penny launched travels the farthest while the maximum height that it achieves is kept at a minimum.