Day 60: The True Meaning of Horizontal Transformations

We’ve spent a few days working on transformations on sinusoidal functions, particularly focusing on the horizontal transformations.  I don’t want the kiddos to simply say that we “do the opposite,” but to really understand the idea behind what we see in the equation versus what is physically happening to the graph.

At our school we use the term “point-rule” to describe what is happening to the function.  This convention becomes very helpful in discussing what is really happening.  Here’s a snippet of our class discussion:

But how can we get an equation for the transformed graph?   First, we let’s consider the point rule relationship.  We take a parent’ input (let’s call it xP …this is subscripted but I can’t show that in this free blog format…sorry)), add  π/2, and get the transformed’s input, xT.  Symbolically, we can say xP+π/2 = xT .  Since the sine relationship, output = sin(parent input), requires a parent input, let’s re-write the point-rule relationship: xP+π/2 = xT as xT – π/2 = xP.  This gives an expression for the parent input based on the transformed x-value.  Thus, y = sin(parent input) = sin( xT – π/2) = sin( x – π/2).

Lots of great clarifying questions during the period, but I believe they are really getting it!  What are tough concepts you teach that are worth spending time on for the long run benefits?