A Classroom Note on the Vees of a Function

(ProQuest: ... denotes formulae omitted.) INTRODUCTION While teaching at various levels, the authors have encountered a few students each year who, when confronted with the task of finding the zeros of a quadratic function, bring the constant to the right-hand side of the equation and the other terms to the left. They end up with ax^sup 2^ +bx = -c . What happens next is never pretty. Sometimes they factor, sometimes they divide, and sometimes they stare in bewilderment, but seldom do they find the zeros of the function. Of course these students have been taught how to handle quadratic equations, but they lapse into the incorrect, but more familiar, approach used to solve linear equations. The following paper is based upon our discussions of the question we would like our students to ask more often, "Why should it be so?" We provide an alternative approach to the solutions of linear and quadratic equations for the consideration of teachers of elementary algebra. Students who are simply looking for the answer to the problem of the moment may be reverting back to an algorithm for linear equations. This process reduces linear equations to .... Their frequent success with this process encourages students to file this method into their mathematical toolbox and draw upon it often. After all, most students have little or no trouble with a problem such as: If the cost of a cab ride in Smalltown is 3x + 2 , where ? is number of miles traveled, how many miles can a person ride a cab for $20. Students simply solve: ... Of course, students have been introduced to the quadratic formula that never fails to find all zeros, and thus all roots of the quadratic: integers, real numbers, and even complex numbers. Perhaps, it is the complex number possibility of the quadratic equation that brings many to the linear option first. UNIFYING THE PRELIMINARY APPROACH The process of preparing both linear and quadratic equations for solution could be identical, until the final step. This approach prevents those anxious to get "the answer" from mistakenly identifying an algebraic equation as either linear or quadratic before the preliminary simplification. For example, a student who encounters x(x + 7) = x^sub 2^ + 18 may approach it as quadratic, and force this belief on the method of solution. By using a uniform formulaic approach for both linear and quadratic equations, activities such as removing parentheses, adding or subtracting the same quantity to both sides of the equation in sufficient steps to bring all "x-terms" to one side, and simplifying, all have a common purpose: Find a function equal to a constant. The task becomes to determine the choice of ? for which the value of a function is equal to that constant, whether it is zero or not. We call these values of* the "Vees" of the function. At this stage, the appropriate formula, linear or quadratic, is brought to bear. Of course, the two formulas are different. The Linear Case: In the case of linear equations such as 3(x-2) = x + 4 and using the uniform approach, the task is to find the choices of ? for which any function in the form f(x) = ax + b has a value of V regardless of whether V is zero or not. Then ax = V - b yields a formula we call the Linear Vees Formula .... The Quadratic Case: This "Vees" approach can also be used to solve a quadratic equation such as, ax^sup 2^ + bx + c = V a quadratic function equal to a constant. The development below finds the choices of jc for which the rule y = ax^sup 2^ +bx + c has a y-value of V. It does not matter if one, the other, neither, or both of c and V are equal to zero. The appropriate formula can be derived by completing the square: The Vees of a quadratic function, which might or might not be real, are the solutions of the equation ax^sup 2^ + bx + c = V given by The Quadratic Vees Formula. ... This approach sets the stage for some interesting classroom discussions about the associated graphical solutions, and/or the existence of a cubic formula and the approaches to finding the zeros of the quartic polynomials using transformations. …