Transformations are an important topic when learning how to graph a function.
This is because with transformations you only need to memorize a handful of basic operations.
With these operations you can then stretch, flip, or shift a function into its proper
shape.
Now when you learn all of the various transformations you quickly learn that they can fall into
two categories based on how they effect the parent function.
There are the inside transformations, and outside the transformations
These of course get their names because they can be found either on the inside or the outside
of the parent function.
When it comes to the outside transformations these effect the outputs or the y-values of
a function and they behave much like you would expect.
If we multiply a function by 5, it stretchs the function by 5
Add 3 to the outside, it goes up by 3
But now comes the interesting part. The inside transformations seem to work in
the opposite way you would expect. They effect the x-values so we see the result
in the horizontal direction, but watch closer. multiply the inside by 5, and it will shink
or compress the function by 5. Add 3 to the inside, and it will move to the
left 3 units
So why does this happen? Well the truth is that these inside transformation
are not really backwards at all.
To convince you that these transformations are working normally
Let's play around with a function and watch what happens when we put in a value for x,
say x=2.
First we have 2 - 3 giving us a result -1. Next we multiply this by 2 and square it.
We can then multiply by 4 and add 1 giving us a final result of 17.
The entire time we apply the operations, nothing is opposite in any way.
So then why is it when we go to graph the function we then have to apply in the insdie
transformations in the opposite way?
To get to the heart of this question we are going to need to look at the function in a
different way. Rather than say inside or outside transformations,
we'll break the process into individual functions. This way we can better keep track the inputs
and outputs of each transformation. In this diagram every transformation is its
own mini function.
From this you can see that the inside transformations are really just transformations that come
before the parent function, and the ouside transformations are the ones that come after
the parent function.
Using this diagram we can see what happens when we put in an input like x=2, and follow
it through the transformations. As expected nothing unusual happens, because
afterall these transformations are not backwards.
Now when we are graphing, we are not really starting at the begining of this chain of
inputs and outputs. Instead we are starting in the middle with
the parent function.
If we had a value at this point, all of the transformations that come after it would effect
it as usual. But we would have to work backwards to figure
out exactly what original input this value came from.
This of course involves working each of our tranformations backwards as well.
So the reason why it seems we are doing the inside transformations "backwards" is because
we are reading them in relation to the parent function.
In many ways we are using the parent function as a frame of reference.
Everything that happens after it works normal, but we have to read opperations in the opposite
way when searching for the inputs that came before it.
In this proces the parent function is usually a great place to put our frame of reference.
It usually has a distinctive shape or equation, but this process reveals that our frame of
reference is somewhat arbitrary.
If we want we can choose a different frame of reference. By applying what we just learned,
we should still be able to create the correct graph.
We'll just have to be extra careful that we are reading our transformation correctly from
our choosen frame.
Let's give this a try. For this part we'll use the following function.
The first way interpret this is the usual way with the parent function as y = x^2.
This would make the 2 and -3 all outside transformations and the +1 an inside transformation.
Now when graphing we'd be sure to stretch by a factor of 2 and shift down 3.
Since +1 is on the inside, or comes before the parent function, we'll work backwards
to find the proper x-values. Visually we'll see that the graph shift to
the left by 1.
Now a second way is to start with the parent function of y = x.
In this interpretation the +1, squaring, 2 and -3 are all outside transformations.
So let's start with our parent function y = x and see what happens in this context.
First we'd shift it up by 1 Apply a squaring transformation
Stretch it vertically by 2 and lastly move it down by 3
Note how we have exactly the same graph as before!
So in summary, when we talk about outside and inside transformations, these are really
in reference to some given function. The inside transformations seem to be backwards,
but this is only becasue they are applied before our given reference point.
We are the ones who are working backwards through them to find the proper inputs.
Most importantly, we can change our frame of reference, and hence change what gets to
be called an outside or inside transformations. As long as we still read them in the correct
way, we will still come up with the correct graph every single time.
Hopefully that sheds some light on this useful technique for graphing and makes you a little
bit more curious of the techniques we actually use.
Next time you see your teacher, don't forget to ask them why inside transformations seem
to work backwards and see what they have to say.
Thanks for watching.
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