ISSN: 2250-3005 - ijcer

International Journal Of Computational Engineering Research (ijceronline.com) Vol. 2Issue. 8A New Geometric Method to Plotting a Familyof Functions Logarithm1, B. Nachit, 2, A. Namir, 3, M. Bahra, 4, K. Hattaf, 5, R. Kasour, 6, M. Talbi1,2 Laboratoire de Technologie de l’Information et Modélisation(LTIM), Faculté des Sciences Ben M’Sik, Universit éHassan II-Mohammedia, Casablanca, Maroc1,3,5 Cellule d’Observation et de Recherche en Enseignement des Sciences et Techniques (COREST), Centre Régionaldes Métiers de l’Education et de la Formation Derb Ghalef, Casablanca, Maroc4Département de mathématiques et informatiques, Faculté des Sciences Ben M’Sik, Université Hassan II-Mohammedia,Casablanca, Maroc1,5,6 Observatoire de Recherches en Didactique et Pédagogie Universitaire (ORDIPU), Faculté des Sciences Ben M’Sik,Université Hassan II-Mohammedia, Casablanca, MarocAbstract:In this paper, from the study of the family of logarithmic function, we derive a new method to construct thecurves:y= kx+ln(x), kIR. This method will be a new learning situation to present the logarithm function at high school.Keywords: Algorithm, Family of functions, Function, Logarithm, Register1. IntroductionThe visualization is very important in the teaching of analysis [8]. The notion of function as an object ofanalysis can intervene with many frames [4] and it is related to other objects (real numbers, numerical sequences ...).This concept also requires the use of multiple registers [5], that are, algebraic Register (representation by formulas);numerical register (table of values), graphical register (curves); symbolic register (table of variations); formal register(notation f, f (x), fog ...) and geometrical register (geometrical variables).In addition, Balacheff and Garden [1] havefounded two types of image conception among pairs of students at high school, that are, conception curve-algebraic, i.e.,functions are seen as particular cases of curves, and a conception algebraic-curve, i.e., functions are first algebraicformulas and will be translated into a curve.The authors Coppe et al. [3] showed that students had more difficulties totranslate the table of variations from one function to a graphical representation which shows that students havedifficulties to adopt a global point of view about the functions. They have also shown that the algebraic register ispredominant in textbooks of the final year at high school. They also noted that the study of functions is based on thealgebraic calculation at the final year in high school (limits, derivatives, study of variations...). According to Raftopoulosand Portides [7], the graphical representations make use of point of global and punctual point of view of functions; onthe contrary, the properties of the functions are not directly visible from the algebraic formulas. Bloch [2] highlightedthat students rarely consider the power of the graphics at the global level and propose teaching sequences supported by aglobal point of view of the graphical register. The students do not know how to manipulate the functions that are notgiven by their algebraic representations. And they do not have the opportunity to manipulate the families of functionsdepending on a parameter.To study the logarithm three methods are available:a- From the properties of exponential functions.b- Put the problem of derivable functions on IR +* such as f (xy) = f (x)+f (y) and admit the existence of primitive for thefunction x 1/x (x≠0).c- Treat the log after the integration. In this paper, we propose a new method of tracing the family of functions f k (x) = kx+ ln(x), k IR, without going through the study of functions (boundary limits, table of variation s, infinite branches ...)based only on algorithms for tracing tangents.This method will be used in particular to plot the curve of the logarithm function and the student can from the graphicalrepresentation find the properties such as domain of definition, limits, monotony… etc. The idea of this new method isbased on our work presented in [6].2. Description of the methodLet f k be a family of functions defined the interval ]0, +∞[ byf k (x) = kx + ln(x)With k IR. The function f k is strictly increasing on ]0, +∞[ when k ≥ 0. If k < 0, then f k is strictly increasing on]0, [ and is strictly decreasing on ] 1 ,+[. Moreover, the line y = kx is an asymptotic direction of graph of fk , andkk 1the equation of the tangent at any arbitrary point x 0 isIssn 2250-3005(online) December| 2012 Page 96