Nowadays, with the help of a computer or graphing calculator, we can easily obtain an accurate graph of a function if we are given the formula that represents it. However, suppose that we are given the graph of a function and asked to determine the parameters of the formula that generated it. We would like to share one method of finding the coefficients of the quadratic function. The method may differ from methods that readers have previously seen (Metz 1994). Our approach describes a property of the parabola that is connected with the main coefficient a in the standard representation y = ax2 + bx + c.
Although we no longer doubt the legitimacy of split domains as a function, students still have difficulty perceiving a graph having a shape like that of Euler and d'Alembert's string as the graph of a single function rather than of several joined functions (Vinner 1989). Given the centrality of the function concept in the school curriculum (NCTM 1989; Amit and Koren 1993), one cannot underestimate the importance of exposing students to this type of function. A method of introducing students to this concept while furnishing the teacher with material that has its own intrinsic interest (Satianov 1995).
In studying algebra, a segment of the number line is inevitably related to a system of inequalities in one variable. Similarly, a convex region of the plane is inevitably related to a system of inequalities in two variables. Indeed, we drew this unsurprising conclusion from a questionnaire that we gave to a group of high school teachers and a large group of advanced twelfth graders. In the questionnaire, we simply asked what kind of mathematical expression is needed to determine the points in a line segment and in a triangular region.
Teachers of first-year college mathematics and engineering courses must often spend considerable time reviewing material originally taught in high school. Instead of this being a mere exercise in repetition, this article suggests that such a review can enrich and revitalize by unifying some of the subjects that need to be re-taught. In the example presented, the subjects in question are absolute values, graphs and solutions of equations, and domains of definition. These are unified by the problem of finding an analytic expression for a square and triangle and their interiors. In the course of the development, basic notions such as the additive property of areas and convexity are introduced. The approach presented in the article was tried with secondary school teachers participating in professional development workshops and with students at a technical college; the teachers and students responded enthusiastically to the material.
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