In this lesson, the instructor will focus on the use of the following quadratic model to solve equations in which at least two points on a parabola are given: ax2 + bx + c. It is also a given that the parabola in question has a vertical axis and is symmetrical. The lecture and in-class examples (which the students will be expected to solve) will focus on demonstrating how one can determine the coefficients of 2nd degree polynomial equations.
In this lesson, students will:
- Listen to a lecture about quadratic models, the ax + bx + c model in particular.
- Learn how to solve quadratic polynomial equations.
- Review the following vocabulary words: coefficients, polynomial equations, parabola, quadratic model, symmetry, overdetermined equation, general form.
- Work on in-class examples provided by the instructor and provide answers orally and/or by handing in their papers or worksheets.
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?
This post is part of the series: Learning About Graphs
- A Lesson on Graphing Equations Using the Graphing Calculator
- Teaching About the Rectangular Coordinate System
- Teaching the Zero Method For Solving Graphic Equations
- Lesson plan: Solving Graphic Equations with the Intersection Method
- Lesson Plan for Teaching Students to Solve Polynomial Equations