id: 02310460
dt: j
an: 1996a.00471
au: Ligh, S.; Wills, R.
ti: On linear functions II.
so: Math. Comput. Educ. 29, No. 1, 32-52 (1995).
py: 1995
pu: MATYC Journal, Old Bethpage, NY
la: EN
cc: G74
ut:
ci:
li:
ab: Let L(R) be the set of linear functions defined over the real number R.
Once we have characterized the subset of L(${\bbfR}$) corresponding to
the subset of ${\bbfR}^2$ generated by a function y = f(x) to be the
set of lines tangent to the graph of y = g($χ$) we will show how to
use this result to obtain a whole family of related results. We do this
by demonstrating what effect translations and dilations of the function
y = f($χ$) have on the function y = g($χ$). It can be shown that any
quadratic function can be expressed in terms of translations and
dilations of y = $χ^2$, the upper and lower halves of any non-rotated
ellipse can be expressed in terms of translations and dilations of y =
$\sqrt{1-χ^2}$, and the branches of any nonrotated hyperbola can be
expressed as translations and dilations of either y = $\sqrt{1+χ^2}$
or y = $\sqrt{χ^2-1}$. Thus, our theorem will enable us to
characterize the subsets of L(${\bbfR}$) corresponding to the subsets
of ${\bbfR}^2$ generated by functions whose graphs are conic sections
by knowing the subsets of L(${\bbfR}$) corresponding to the subsets of
${\bbfR}^2$ generated by the four functions y = $χ^2$, y =
$\sqrt{1-χ^2}$, y = $\sqrt{1+χ^2}$, and y = $\sqrt{χ^2-1}$. The
software package Maple is used to illustrate some examples. (orig.)
rv: