1. Transformations of f(x)

So, I am working with my son on transformations of functions.
He is ok with stretches and translations in the y direction but when we look at f(ax) or f(x+a) where the 'opposite' seems to happen to what he expects he gets lost! Anyone got a good way of thinking about this that can help him understand what is happening and why?
What is the best function to pick? I realised f(x) = x^2 is confusing!

2. Re: Transformations of f(x)

Let's compare f(ax) to f(x). Let's compare input to output:

$\begin{matrix}x & | & f(x) & | & f(2x) \\ -- & | & ---- & | & ---- \\ -5 & | & f(-5) & | & f(-10) \\ -3 & | & f(-3) & | & f(-6) \\ -1 & | & f(-1) & | & f(-2) \\ 0 & | & f(0) & | & f(0) \\ 1 & | & f(1) & | & f(2) \\ 3 & | & f(3) & | & f(6) \\ 5 & | & f(5) & | & f(10)\end{matrix}$.

So, when $a=2$, you get a compression of the graph by a factor of 1/2 because you only need half the value of $x$ to get the same input to the function. If you want the input to the function to be $-5$, you need to use the x-value of $-2.5$ because $-2.5\cdot 2 = -5$.

Similarly create a chart for $f(x+a)$. Consider what happens when $a > 0$. What needs to happen to $x$ to get the same input to $f$?

Next, consider what happens when $a<1$ for $f(ax)$ or when $a<0$ for $f(x+a)$.

That is how it helped me to think about it.

3. Re: Transformations of f(x)

I see, suppose i have y= f(x) and f(a)=k then if i consider y= f(2x) , then x would have to be 0.5a in order to get same output k, i.e. f(2(0.5a))= f(a)= k
It's subtle because we now need to shift our thinking to ' what input gives the same output' rather ' how has the output changed?'
Am i making a meal out of this?

4. Re: Transformations of f(x)

Originally Posted by rodders
I see, suppose i have y= f(x) and f(a)=k then if i consider y= f(2x) , then x would have to be 0.5a in order to get same output k, i.e. f(2(0.5a))= f(a)= k
It's subtle because we now need to shift our thinking to ' what input gives the same output' rather ' how has the output changed?'
Am i making a meal out of this?
Exactly!

5. Re: Transformations of f(x)

always set your equations to zero to find the answers, i.e, when you have an f(x-2) per say, we take the x-2=0. from there we shift the two to the right side and we can see that the value represented is indeed a positive number (x=2). As for the parabolic function f(x)=x^2, we need to understand that the coefficient in front of the x value dictates the direction of which the parabola will open, in this case it'll open upwards and look like a U. Also, having it not have a k value, the vertex is (0,0). To prove this, we can use our f(x)=a(x-h)^2+k. where h is the point of the vertex on the x axis, and k is the vertex point on the y. Now to conclude this with the transformations issue we were having earlier, we can use f(x)=a(x-0)^2+0. We are given that a= 1 so we don't need to calculate that. Final answer is f(x)=x^2 Hope that helps

6. Re: Transformations of f(x)

High school math teacher here ... here are my thoughts on the learning processes of this topic ...

It is a very important and an admirable goal to try to get every student to comprehend this topic, but sometimes initially the best route is to train the brain and mathematical reflexes to do something by instinct, rather than a comprehensive understanding of the material. In this way, you create a base of experience and knowledge that can be gently challenged as the learner deepens his or her understanding of the rules. So, I suggest that you:

1) Have your son memorize that actions on the variable, x, will do the opposite of what it "looks" like (i.e. 2x, is a horizontal compression by a factor of 1/2) and actions outside of the function (that act on the output, y) will do what is expected
2) Given enough time, examples, and success, if time permits, pursue a deeper understanding of why this occurs (analyzing the expressions algebraically, input vs. output etc.)

Your son is very lucky to have your support and effort on this!

Marian

7. Re: Transformations of f(x)

Very interesting perspective. And i think sometimes this is the right way. Otherwise, as i am discovering, frustration sets in too early and kids give up.