Describing Graph Transformations with Input / Output diagrams

I regularly stress to my grade 10 and 11 students that when transforming a function it is imperative to dilate before translating in any direction. In other words, if a function is both stretched and translated horizontally the stretch must happen first. Ron Watkins’ demonstration of Input / Output diagrams at OAME2014 has helped to see how rearranging the equation that describes a function can translate to rearranging the kind and order of transformations

An Input / Output diagram shows the sequence of operations applied on a function’s argument, x, to arrive at its value g(x). Naturally this sequence follows order of operations. What is interesting is that transformations are made clear through the diagram. Look first at the parent function indicated by the red arrow, which in this case is f(x) = x². Moving towards the right are the vertical transformations: stretch by factor 3 then translate 5 up. Following the arrows towards the left are the horizontal transformations: compress by factor one half, translate 4 right.

 Often students don’t see why horizontal transformations on the parent function are related to the inverses of operations seen in the function’s equation.  The diagram shows that to go from the input of the parent function (red arrow) to the horizontal coordinate x requires inverse operations, while going from the output of the parent function to g(x) does not.



Rearranging the equation for g(x) by distributing the 2 gives a function, h(x), where order of operations leads to rearranging steps in the Input / Output diagram, and so also a related change in the order of transformations to the graph of the parent function.

Algebraically we know that h(x) and g(x) are equivalent. The diagram correctly shows that horizontally, a translation 8 right followed by dilation by factor one half is equivalent to a dilation by factor one half followed by translation 4 right ... a nice way to understand equivalence of graphical transformations. Now, I think it still may be helpful to ask students to factor out the 2 to isolate the effects of different parameters in the equation, but understanding alternative ways to transform could add depth of understanding. Next, could students describe the sequence of transformations in j(x) shown below? With an Input / Output diagram I bet they could.