Trying to understand function transformations

In summary, the conversation revolves around understanding function transformations and whether they are axioms or can be proven. The example of shifting a function 3 units up and to the left is discussed, with the explanation that translation can be given by adding and dilation can be given by multiplying. The process of proving these properties is briefly mentioned, but it requires a strong background in geometry.
  • #1
pctopgs
20
0
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
 
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  • #2
So what's your question?
 
  • #3
Transformations are pretty self-explanatory . . .
TransformationsOfFunctions_4_pg1.png
 
  • #4
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?
 
  • #5
pctopgs said:
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.
 
  • #6
pctopgs said:
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##.
 
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  • #7
@micromass even after taking a course entirely focused on signals, that was probably the clearest explanation I've seen for function shifting.
 
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  • #8
axmls said:
@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.
 

1. What are function transformations?

Function transformations refer to the changes that occur to a function when certain mathematical operations are applied to it. These can include shifts, reflections, stretches, and compressions.

2. What is the purpose of studying function transformations?

Studying function transformations can help us understand how changes in the input of a function affect its output. This is important in many fields such as physics, economics, and engineering where functions are used to model real-world phenomena.

3. How do shifts affect a function's graph?

Shifts, also known as translations, move the entire graph of a function horizontally or vertically. A shift to the right by a certain amount will result in the graph moving in the positive direction on the x-axis, while a shift to the left will result in a negative direction. Similarly, a vertical shift up will result in a positive direction on the y-axis, and a shift down will result in a negative direction.

4. What is the difference between a reflection and a stretch/compression?

A reflection is a transformation that results in a mirror image of the original function. This can occur over the x-axis, y-axis, or a line other than the x or y-axis. A stretch or compression, on the other hand, changes the shape of the graph without changing its orientation. A stretch will result in a taller or wider graph, while a compression will result in a shorter or narrower graph.

5. How can I identify function transformations from an equation?

To identify function transformations from an equation, you can look for specific patterns. For example, a term added or subtracted inside the parentheses will result in a horizontal shift, a coefficient outside the parentheses will result in a stretch or compression, and a negative sign outside the parentheses will result in a reflection. Additionally, the powers of the variables can also indicate stretches or compressions.

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