There are many different types of transformations we can apply to functions and their corresponding graphs. Think about any image processing programs you have ever used where you could crop, skew, rotate, move or otherwise manipulate an image. Many of those types of transformations have a direct correspondence to the function transformations we will be studying.
When performing transformations on a given function, say \(f(x)\), we will typically write \(y=...\) to indicate a new graph based on the old function after applying one or more transformations. Typically, operations that occur outside the original function result in vertical transformations while those occuring inside the function parenthesis are horizontal transformations.
Shifting a Function
The first transformations we will consider are horizontal and vertical shifts (also called translations). Consider some positive real number values \(h\) and \(k\) and a function \(f(x)\) defined on real numbers.
Note that the horizontal shifts are the opposite sign to which you might expect.
Next up are the stretching or shrinking transformations. Let \(a\gt 1\) be a real number.
\[\begin{{array}}{{rl}} \text{Horizontal Stretch by a factor of }a:&y=f\left(\frac{{x}}{{a}}\right)\\ \text{Horizontal Compression by a factor of }\displaystyle\frac{{1}}{{a}}:&y=f(ax)\\ \text{Vertical Stretch by a factor of }a:&y=af(x)\\ \text{Vertical Compression by a factor of }\displaystyle\frac{{1}}{{a}}:&y=\displaystyle\frac{{1}}{{a}}f(x) \end{{array}} \]Once again, note that the horizontal operations of dividing and multiplying correspond to opposite transformations of stretching and compression!
The last set of transformations are similar to stretching and compression, but instead are about flipping or reflecting the graph over a specific axis:
\[\begin{{array}}{{rl}} \text{Reflection over the }x\text{-axis}:&y=-f(x)\\ \text{Reflection over the }y\text{-axis}:&y=f(-x) \end{{array}} \]Note that rotation over the horizontal, \(x\), axis is a vertical transformation, so the negative is outside the function.
The following graph is a function and the applied transformation(s): \(y = a f(x-h)+k\). You can slide the points to pick different values of \(a,\;h,\;k\) and see what happens.
For another perspective, here we are applying the transformation(s) \(y = f(a(x-h))+k\). Notice how \(a\) affects this graph differently than the previous one.