Here’s a shameful post – one of those things I knew it happened, and wanted to believe I knew WHY it happened, but in reality…I was struggling. Until yesterday… in after school, tutoring a student for upcoming comprehensive final.

I know how manipulating an equation will transform the graph of the function. I can predict it quite efficiently. I know and my students even recognize that y=a (x-h)^2 +k will translate left / right… “opposite” of what the sign / operation is in the parentheses with the h. But why?

So one day, as I heard myself describing the transformation to a student, I thought…that sounds so stupid. I mean, hello. No wonder it doesn’t stick. It makes no sense (what I had just said). In my mind, I heard Pam, the student, asking why do we change the sign of the h, but not the k? Why does the h move opposite, but the k translates direction of the operation?

I started really making more sense to myself when I focused on function transformations in Algebra 2 and transformations for plane figures in Geometry the past couple of years. But I was definitely not satisfied with what I was saying. I believe our work with equations of circles related to slope and pythagorean theorem is what started chipping away my lack of true understanding. Because I began to explore, ask questions. I was curious.

When I started having students create tables of values, seeing how the values changed with each transformation helped, but not to the level I’d like.

So, here it is folks… when we’re looking at the y=a(x-h)^2 + k…the h is actually NOT the x-coordinate of the vertex. The h is the transformation back to zero (origin). Can we look at it that way? Does that even make sense? The x-value is where we moved from the origin. The h will return us back to the origin. I know its not where I need to be yet. But I’m open to listening to other’s ideas here. I’m not satisfied with “it moves just opposite of what we think.”

My next failure as a teacher saga…I don’t do a good job of helping students differentiate between linear functions and arithmetic sequences. I’m starting to muddle an understanding. At a moment in time, they are comfortable with each idea, but they continue to mix up when its a first term, n=1 OR an initial value n=0. The best I can do for now, verify your equation works for the values…

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I like to think of transformations as what value of x makes the parentheses = 0. Then if the part of the function that relates to x goes to zero, what does y become? (k!) I guess you could also start out with the idea that the parent function has a center or vertex or whatever at (0, 0) so h and k are 0 and 0. That way the transformation is built in from the beginning, but it does tend to make our equation a lot more congested.

Here’s an analogy I’ve used with the kids… It’s like a see-saw. When you do something, it transforms the graph in the OPPOSITE way. So if you have (x+3)^2, then the graph will shift horizontally more negative (-3) to counter-act the +3.

Now, why does the stuff inside work different than the stuff outside? It actually doesn’t. If you move the stuff outside to be “next to” the thing it’s acting upon, there is no inconsistency! So if you have y=(x+3)^2 -5 you could rewrite as y+5=(x+3)^2 and then it’s obvious that they actually work the same.

Goes for stretches, not just shifts, too! And I like to show them stuff like y=(2x)^2 is a horizontal shrink by a factor of 2, but that’s the same as y=4x^2 which is a vertical stretch by a factor of 4. So those two transformations are equivalent. Which should make some intuitive sense to them, hopefully.