Quadratic equations: Graphical and analytical relationships
69 QEGraph1 A quadratic equation is presented. The student moves the cursor to various locations on the X axis and presses enter. The calculator evaluates the quadratic and displays the corresponding value for Y. By trial and error, the student locates the two values of X that make the quadratic expression equal to 0. . A tolerance of + or - 0.5 is allowed in reading the X value of the point The purpose is to introduce the student to the nature of the graph of a quadratic and to the notion that the solutions to ax²+bx +c = 0 are the points at which the parabola ax²+bx +c = y crosses the x axis. It is probable that this program and the next, QEInequa, could be done in a single 90-minute session by a sharp class.
70 QEInequa Presented with a quadratic inequality, the student must locate any two points on the corresponding parabola and then place a point in the solution region. A tolerance of + or - 0.2 is allowed in placing the Y value of the point You may want to suggest that the students find the vertex ( Xv= -b/(2a), Yv=AXv²+ BXv +C ). Of course, the easiest points are found by letting X be 0 and 1 or -1.
71 QESqRoot This program is the introduction to analytical solutions to quadratic equations. If there is no first power term in the equation, then the analytical solution is simply a matter of taking a square root If the equation presented has no real solutions, the response should be the letter N.
72 QEFactor The notion of the special property of zero products is presented and used to solve quadratic equations which are readily factorable.
73 QECmplSq The method of solution by completing the square is presented and exercised, not because of its utility in solving quadratics, but as a stepping stone to developing a general solution, the quadratic formula. In most of the problems, students are required to input the equation which shows explicitly the square of the binomial. This is an attempt to counter the student's tendency to memorize a meaningless process for obtaining an answer. The program requires exact answers (use Ö ).
74 QEDerive This program prompts the student in ways which enable him/her to derive the quadratic formula from the standard form of the quadratic equation, using the method of completing the square. The program does not keep a record of the number of mistakes or false moves made. It simply congratulates the student on successful completion of the task. Even the better classes will probably need to have a lecture type walk-through because of the novelty of completing the square, taking a square root, and rearranging literal fractions.
75 QEFormul The program provides practice in using the quadratic formula. Students should be cautioned that -3 ² is not equal to (-3) ². The program requires exact answers, which means that, in most cases, the Ö symbol will have to be used.
76 QEGraph2 In this program the student must locate the vertex and both roots of the given quadratic equations, if they exist.
77 QEWordP1 This word problem deals with maximizing the size of an advertising billboard, given certain cost and form constraints. It is fairly difficult for beginning algebra students, so they should be told that the program will not keep track of errors, only successful completion of the task. Prompts are provided..
78 QEWordP2 This program presents a single problem, the time taken for a rifle bullet, fired straight up, to hit the ground (ignoring wind resistance). The student is prompted in the steps to the solution. No record is kept of mistakes. This program is a good preparation for the following one, QEDiscrm.
79 QEDiscrm Some problems involving quadratics can be solved using only the value of the discriminant. This program demonstrates the ideas using the relationships of free-fall. Even the better algebra students may find these problems challenging. You may want to guide the class through the first four problems in lock-step.