Graphing Functions using Transformations

“ A picture is worth a thousand words ” is a well known proverb. To know about a function well, its graph will help us more than its analytical expression. To draw graphs quickly without plotting many points is an invaluable skill. Familiarity with shapes of some basic functions will help to graph other complicated functions.

Understanding and usage of symmetry and transformations will then enable to strengthen graphing abilities. This section is not simply a data base of graphs, we learn some methods to graph certain functions. Suppose that we want to draw or sketch the curve of the function y = sin( x − )+ . At the very first sight it looks that it is very difficult to draw the curve representing this function.

But it will be very easy to draw after understanding the content of this section. If we know a half of a graph is the mirror image of the other half with respective to a line, or a graph can be obtained just by moving a known graph in some direction, then we can draw the new one using the known one. Moreover if we know that a graph can be obtained by enlarging or shrinking a known one, then also we can draw the new one using the known ones. The following type of transformations play very important roles in graphing.

(i) Reflection (ii) Translation (iii) Dilations. . Graphing Functions using Transformations In the case of reflections and translations, they produce graphs congruent to the original graph; that is, the size and the shape of the graph does not change, but in dilation, it produce graphs with shapes related to those of the original graph. Reflection The reflection of the graph of a function with respect to a line ℓ is the graph that is symmetric to it with respect to ℓ .

A reflection is the mirror image of the graph where line ℓ is the mirror of the