id: 06649819
dt: j
an: 2016f.01130
au: Miroshnikova, Natalia N.
ti: Teaching congruent transformations of graphs of power and exponential
functions.
so: Far East J. Math. Educ. 16, No. 2, 203-219 (2016).
py: 2016
pu: Pushpa Publishing House, Allahabad, Uttar Pradesh, India
la: EN
cc: I20
ut: method of invariant points; power and exponential functions; curve
sketching
ci:
li: http://www.pphmj.com/abstract/9771.htm
ab: 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.
rv: