Trying to understand function transformations

[pctopgs]

Hey guys,

This isn't a homework question but i learned about it in school. I'm trying to gain a more fundamental understanding or function transformations.

that's the typical parent function for example:

f(x)=x^2

I would move the function 3 units up, I would write it lil this:

g(x)=x^2+3

 

[ProfuselyQuarky]

So what's your question?

 

[ProfuselyQuarky]

Transformations are pretty self-explanatory . . .

 

[pctopgs]

Sorry I'm having some technical difficulties posting.

I know about the transformations, but I wanted to ask if these are axioms or if they can be proven.

f(x)= x^2+3 moves the graph 3 units up. This makes sense be caus f(x)=y, but why does putting parentheses around 'x^2+3' magically make the function move 3 units to the to the left instead?

I'm asking if these are just rules that we just have to accept, or are these rule derived from something? If so, what is it?

 

[PeroK]

Sorry I'm having some technical difficulties posting.

I know about the transformations, but I wanted to ask if these are axioms or if they can be proven.

f(x)= x^2+3 moves the graph 3 units up. This makes sense be caus f(x)=y, but why does putting parentheses around 'x^2+3' magically make the function move 3 units to the to the left instead?

I'm asking if these are just rules that we just have to accept, or are these rule derived from something? If so, what is it?

Do you mean $g(x) = f(x+3) = (x+3)^2$?

Which means that $g$ is $f$ moved 3 units to the left.

 

[micromass]

Sorry I'm having some technical difficulties posting.

I know about the transformations, but I wanted to ask if these are axioms or if they can be proven.

f(x)= x^2+3 moves the graph 3 units up. This makes sense be caus f(x)=y, but why does putting parentheses around 'x^2+3' magically make the function move 3 units to the to the left instead?

I'm asking if these are just rules that we just have to accept, or are these rule derived from something? If so, what is it?

Oh no, they most definitely can be proven. But all really depends on your background. You'll need to know some geometry.

Let me pick one. Translation in general can be given by adding. So if we translate a point $\mathbf{x}$ by a point $\mathbf{v}$, then the translation is given by $\mathbf{x} + \mathbf{v}$ (this rule can be proven too, but requires some more deeper geometry). In particular, a horizontal translation of length three means that we translate our point $\mathbf{x}$ parallel to the X-axis with length $3$. Clearly, this kind of translation is given by $(x,y) \rightarrow (x,y) + (3,0)$.

Now let's take a function $f(x) = x^2$. The graph of this function consists of a collection of points, namely the points $(x,f(x))$, so $(x,x^2)$. Translating the graph horizontally with length three means translating every point in the graph. An arbitrary point in the graph has the form $(x,x^2)$, which yields after translation $(x,x^2) + (3,0) = (x + 3, x^2)$. So the translated graph consists of all points $(x+3, x^2)$. Said differently, for every $x$, we are given a point in the translated graph as $(x+3,x^2)$. If we put $z = x+3$, then $(x+3,x^2) = (z, (z-3)^2)$. Clearly, the collection of points given by $(x+3,x^2)$ for every $x$ equals the collection of points $(z,(z-3)^2)$ for every $z$. But the collection of points $(z,(z-3)^2)$ for any $z$ can immediately be recognized as the graph of the function $g(z) = (z-3)^2$.

We can thus conclude that if we start from the function $f(x) = x^2$ and translate its graph horizontally with length $3$, then we obtain the graph of the function $g(z) = (z-3)^2$.

 

[axmls]

@micromass even after taking a course entirely focused on signals, that was probably the clearest explanation I've seen for function shifting.

 

[micromass]

@micromass even after taking a course entirely focused on signals, that was probably the clearest explanation I've seen for function shifting.

You can prove the other properties in a similar manner.

It's not a complete answer since I didn't show why $(x,y) \rightarrow (x,y) + (a,b)$ defines a translation, but I guess it's intuitively so. In the same way, you must accept that $(x,y) \rightarrow (ax, ay)$ defines a dilation. You can prove this, but then you need to start from purely geometical axioms and define translation/dilation/reflection in a suitable way. This path takes quite some time.