Students enter Algebra I having experience with transforming lines, rays, triangles, etc., using translations, rotations, reflections, and dilations from Grade 8, Modules 2 and 3. Thus, it is natural to begin a discussion of transformations of functions by transforming graphs of functions—the graph of a function, 𝑓: ℝ → ℝ, is just another geometric figure in the (Cartesian) plane. Students use language such as, “a translation 2 units to the left” or “a vertical stretch by a scale factor of 3,” to describe how the original graph of the function is transformed into the new graph geometrically.
As students apply their Grade 8 geometry skills to the graph of the equation 𝑦 = 𝑓(𝑥), they realize that the translation of the graph to the right by 4 units is given by the graph of the equation 𝑦 = 𝑓(𝑥 − 4). This recognition, in turn, leads to the idea of a transformation of a function (i.e., a new function such that the graph of it is the transformation of the original graph of 𝑦 = 𝑓(𝑥)). In the example described, it is the function given by 𝑔(𝑥) = 𝑓(𝑥 −4) for any real number 𝑥 such that 𝑥 − 4 is in the domain of 𝑓.
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