1) The instructor will begin the class with the following example: A little leaguer hits a home run and the baseball travels for 125 feet before it hits the ground. When the little leaguer hits the ball, the ball is between 2 and 3 feet high. Putting aside air resistance, the baseball's flight can be modelled using a quadratic equation of the form ax2 + bx + c. How would you go about finding the coefficients of such an equation?
2) Before tackling this problem, the instructor will explain how it is necessary to know certain points on the graph the equation creates. S/he will give an example of a vertical parabola and seven points ((-3, 0), (-2, -12), (-1, -16), (0, 12), (1, 0), (2, 20), (3, 48) on the parabola, as an example of an overdetermined equation. While it may seem counter-intuitive, more points are not necessarily better, as overdetermined equations can lead to rounding errors and inequivalent equations modeling the same data.
3) Next, the instructor will demonstrate how to solve the equation with three points ((-2, -12), (-1, -16), (2, 20)) to find the values of a, b, and c in the equation y = ax2 + bx +c. These three points are not particularly efficient, so the instructor will also use the y-intercept (0, -12) to solve for c much more easily and efficiently.
4) The instructor will return to the original problem posed at the beginning of class with a worksheet that the students will be expected to work on during the remainder of the class session and turn it in before they leave. The worksheet will contain these questions:
- Regarding the initial heigh of the ball: What is happening when x=0?
- What are the possible values for y when x=0?
- What are this/these points called (x=0, y=possible values)?
- What is the name of the point on the parabola called where the maximum height is attained?
- What would be the height of the ball when x=125 feet?
- What is the name of this point? (the point when x=125)
- This value of x has a name. What is it called?