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.)