
06649819
j
2016f.01130
Miroshnikova, Natalia N.
Teaching congruent transformations of graphs of power and exponential functions.
Far East J. Math. Educ. 16, No. 2, 203219 (2016).
2016
Pushpa Publishing House, Allahabad, Uttar Pradesh, India
EN
I20
method of invariant points
power and exponential functions
curve sketching
http://www.pphmj.com/abstract/9771.htm
Summary: A novel method of teaching congruent transformations of graphs by means of invariant points is described for power and exponential functions. Primarily, canonical plots of power functions of even and odd degrees, which pass through three invariant points, are presented. Secondly, substitutions reducing given power functions to the canonical ones are calculated. Finally, the reducing substitutions are used to rescale coordinates of the invariant points. Thus, the method of invariant points significantly simplifies curve sketching since it allows to replace plotting of curves with changing labels of the canonical plots. Furthermore, this method considerably clarifies meaning of parameters of horizontal and vertical shifts, dilations in $x$ and $y$directions, and reflections with respect to $x$ and $y$axis. These parameters naturally appear in the process of calculation of the reducing substitutions and the new coordinates of only three invariant points. The method of invariant points may be used for all elementary functions. Application of this method for sketching graphs of exponential functions is also treated. The paper includes both general solutions for professors and typical exercises for students.