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!