Function Transformations Nitty Gritties

I've been racking my brain about how I am going to break down to my Precalc kids the procedures for how to analyze the various transformations involved in a function that looks complicated like g(x) = -3(-0.5x - 4)^3 + 7... Transformations as in, when you compare g(x) against its parent function, f(x) = x^3, what types of shifts and scaling and reflections had resulted in this new function g. This topic is always really difficult for kids, because I am sure that when a regular kid looks at the many numbers inside a complicated transformed equation, all of the number just melt together and seem to be indistinguishable.

Well, I think I've got the big-picture concept finally nailed down to one picture. I hope this is children-safe. It's sort of like an input/output diagram of a function box, but annotated.


So, basically, the way I see it, some transformations happen outside of the parentheses (such as multiplying by -3 and adding 7, in my example function g...see below for color-coding) because -- if you consider PEMDAS -- they occur AFTER the "main" function of cubing has already occurred. At that point, it's too late to be affecting x-values, so you're naturally affecting only y-values and causing vertical changes. And since kids already know that higher coefficient = steeper graph, it shouldn't be difficult for them to figure out that this means -3 does the vertical flip / vertical stretch by scale of 3, and 7 does the vertical shift.


On the contrary, (if you again consider PEMDAS,) transformations that occur inside the parentheses had occurred BEFORE the "main" function of cubing occurred. So that means they were operating only on the x-values and therefore only resulted in horizontal changes. Again, most kids can figure out that a fractional coefficient of 0.5 makes the graph flatter, so it shouldn't be hard to make the leap that 0.5 = 1/2 stretches it horizontally by a factor of 2. (Since horizontal stretching makes a graph flatter, whereas a horizontal compression would have made the graph look skinnier/steeper, and therefore would have had a coefficient greater than 1 on the inside.) And -- keeping with the theme of "inside" operations affecting only the x-values -- the negative sign on the -0.5 necessarily makes the graph flip horizontally across the y-axis.

The only complication that remains to be explained is why the horizontal shift is 8 to the left, rather than something immediately visible inside the equation, such as 4 units. The way I've always thought about it, special things happen at f(0) for most parent functions. In order for you to capture that same special point on the transformed graph, you have to find the special value x that would result in you still evaluating zero in the "main" function (in this case, by the time you reach the actual cubing operation, the stuff inside the cube would need to be zero in order for you to observe the same special behavior). So, by setting -0.5x - 4 = 0 we get x = -8, or our special point (and every other point from the old graph) has apparently shifted 8 units to the left!

...Obviously, this reliance on order-of-operations can be translated (harhar) to other parent functions as well. Here are some examples, with color coding to show which operations come before and after the "main" function operation.



(Of course, my juniors will probably have trouble with simply articulating PEMDAS in abstract algebraic form. Whenever that happens, I make them actually plug a value into x, in order for them to write down all the steps of what operation happened, in which order, before they arrived at the result.)

What do you think? Is this explanation child-proof, or still way too complicated? I don't want to be giving the kids a bunch of blind rules to memorize that they don't understand, especially because there are already so many parent functions that they are keeping track of. I also made a long GeoGebra exploration activity that will cover most of this material without any lecturing on my part, but I'll blog about it only after I figure out how well my kids can absorb the stuff through the scaffolded (but long!) activity. Wish me luck!!