Correlation between shifting graph of a function and shifting the axes

In summary, when shifting a graph of a function, we can either shift the function itself by a vector (h,b) or change the coordinates by the same vector. This results in the same shifted function, but depending on the approach, the equations may differ slightly.
  • #1
JC2000
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1.To shift the graph of a function :

Vertical Shifts : ## y=f(x) +h## where the graph shifts ##k## units up if ##k## is positive and downwards when ##k## is negative.
Horizontal Shifts : ##y=f(x+h)## where the graph shifts to the left by ##h## units when positive and to the right when ##h## is negative.

2.Upon shift the coordinate axes :

##x=X+h## encapsulates the x-coordinate of any point w.r.t original axes (##x##), the coordinate of the point w.r.t 'new' axes (##X##) and the x-coordinate of the new origin w.r.t original axes (##h##). Likewise for the y-coordinate we have ##y=Y+k##.

3.I understand that shifting the axes to the left is the equivalent of shifting the graph of a function to the right.

My Question : Given the above points, I believe there should be some sort of correlation between the two sets of formulas(?).

My Guess : Since ##y=f(x+h)## where shifts the function to the left, this ought to be the same as ##y=f(X)## where ##X=x+h##, yet this does not conform to the derived formula. (Maybe I am making an error by not clearly labelling each variable?)
 
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  • #2
JC2000 said:
My Guess : Since ##y=f(x+h)## where shifts the function to the left, this ought to be the same as ##y=f(X)## where ##X=x+h##, yet this does not conform to the derived formula. (Maybe I am making an error by not clearly labelling each variable?)
This is trivially true. If ##y=f(x+h)## and I substitute ##x+h## by a ##\text{shrubbery}##, then ##y## becomes ##y=f(x+h)=f(shrubbery)## only that your shrubbery is called ##X##.In general it is easiest to write the entire graph as a set: ##y=f(x)## corresponds to the graph
$$
\Gamma_f := \{\,(x,y)\in \mathbb{R}^2\,|\,y=f(x)\,\} = \{\,(x,f(x)\,|\,x\in \mathbb{R}\,\}
$$
Your shifts are now the shift of a graph by a vector ##\vec{v}=(h,b)## which is pointwise:
$$
\Gamma'_f = \Gamma_f + \vec{v} = \{\,(x,y)+(h,b)\in \mathbb{R}^2\,|\,y=f(x)\,\} = \{\,(x+h,f(x)+b\,|\,x\in \mathbb{R}\,\}
$$
and the shifted function is ##F_s(x)=f(x+h)=f(x)+b=y_s## which corresponds to a change in coordinates by ##x'=x+h\, , \,y'=y+b## so that we have $$y'=y+b=f(x)+b=y_s=F_s(x)=f(x+h)=f(x')$$
We can now either consider the shifted function ##y_s=F_s(x)## in the same coordinate system ##(x,y)## or the same function ##y'=f(x')## in a different coordinate system ##(x',y')\,.##
 
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  • #3
Your explanation did make things a lot clearer, could you please clarify a few things though...

fresh_42 said:
##F_s(x)=f(x+h)=f(x)+b=y_s##

Is this the same as ##F_s(x)=y_s=f(x+h)+b##?

fresh_42 said:
or the same function ##y′=f(x′)## in a different coordinate system ##(x′,y′)##.

Wouldn't substituting ##y'=y+b## give ##y=f(x')-b## instead of what should be ##y=f(x')+b##?
 
  • #4
JC2000 said:
Your explanation did make things a lot clearer, could you please clarify a few things though...
There you ask the right one. I'm notoriously bad with those left - right things and coordinate transformations. Shifting a function and shifting the coordinate axis is basically the same so the main difficulty is to distinguish the two. I had tried only to use equations derived from ##\Gamma' = \Gamma + \vec{v}## because I know of my weakness with those things. Let's see if I confused something.

Let's see.

Case 1:
We keep the coordinates ##(x,y)## and only shift the function.
The original function is ##y=f(x)## and ##\Gamma_f = \{\,(x,f(x))\,\}## its graph.
The shifted function is a new function in the same coordinate system. We call it ##F_s(x)=y\,.##
(##y_s## was confusing, as we still have the same ##y##.)

The graph of ##y=F_s(x)## be ##\Gamma'_{F_s}=\{\,(x,F_s(x)\,\}##. Now this is the same graph and function if we write ##\Gamma'_{F_s}=\{\,(x+h,F_s(x+h))\,\}## because we run through all ##x\in \mathbb{R}## and to run through all ##x+h \in \mathbb{R}## doesn't make a difference: the second coordinate is still the evaluation of ##F_s## at the first coordinate. But then
$$
\Gamma' = \Gamma + \vec{v}=\{\,(x+h,f(x)+b)\,\} = \{\,(x+h,F_s(x+h))\,\}
$$
which means ##F_s(x+h)=f(x)+b## or equivalently ##F_s(x)\stackrel{(*)}{=}f(x-h)+b##.

(Seems I made a mistake as ##(*)## is the notation of the shifted function within the same coordinate system.)

Case 2:
Here we keep the function and change the coordinates.
This means there is no second function ##F_s##, only the function ##y=f(x)##. Instead we have a pair of new coordinates: ##x'=x+h## and ##y'=y+b##. So we shifted the old coordinates by the same vector ##\vec{v}=(h,b)## to get the new coordinates: ##(x',y')=(x,y)+\vec{v}=(x+h,y+b)\,.##
Thus we still have ##\Gamma'_f = \{\,(x',y')\,\}= \{\,(x+h,y+b\,\} = \Gamma_f + \vec{v}## only, that the new graph is still from the same function. Hence in the new coordinate system we get
$$
y' = f(x') \Longleftrightarrow y+b=f(x+h) \Longleftrightarrow f(x)+b=f(x+h) \Longleftrightarrow f(x)=f(x+h)-b
$$

(I hope this is now without any confusion of left - right or plus- minus.)
 
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  • #5
Thank you for your painstaking response.

1)Why is ##y_s## ignored in Case 1? Shouldn't the entire function be different?

fresh_42 said:
##\Gamma'_{F_s}=\{\,(x+h,F_s(x+h))\,\}## because we run through all ##x\in \mathbb{R}## and to run through all ##x+h\in \mathbb{R}## doesn't make a difference: the second coordinate is still the evaluation of ##F_s## at the first coordinate.
2) Wouldn't ##x+h## give a different output than ##x## (hence making a difference to the second coordinate)? Or have I misunderstood the statement?

3)The way I look at it is as follows :

Moving the coordinate axes to the right gives : ##x=X+h## or, ##X=x-h##
Which is the same as shifting the function to the left : ##y=f(x+h')## or, ##y=f(X)## where ##h' = -h##. (The opposite signs indicate the fact that movement of axes/function in opposite directions results in the same movement?).

4)Likewise for moving vertically?

Though my explanation(if correct) seems to be very crude compared to how rigorously you draw the correlation.
 
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  • #6
JC2000 said:
Why is ##y_s## ignored in Case 1? Shouldn't the entire function be different?
Yes, and that is why I called it ##F_s(x)##. You can of course set ##y_s:=F_s(x)##. The problem is, that we still are in the ##(x,y)## world, where the variable comes from the horizontal ##x-##axis and the function value is scaled along the vertical ##y-##axis. I thought it would lead to more confusion if we drew the ##y_s-## values on the ##y-##axis. That's why I thought it is better to avoid the ##y_s## notation and only write ##F_s(x)## for it.

JC2000 said:
Wouldn't ##x+h## give a different output than ##x## (hence making a difference to the second coordinate)? Or have I misunderstood the statement?
Sure. But as we consider the entire set of points, the ##x## output will be achieved at ##x-h## by ##(x-h)+h##.
It is ##(x,f(x)) \neq (x+h,f(x+h))## but ##\{\,(x,f(x))\,|\,x \in \mathbb{R}\,\} = \{\,(x+h,f(x+h))\,|\,x\in \mathbb{R}\,\}\,.##
JC2000 said:
Though my explanation (if correct) seems to be very crude compared to how rigorously you draw the correlation.
Yes, it's ok. You can see it in the final equations of my explanations, too. Look at the inverse signs!

Function shifted by ## + (h,b)\, : \, F_s(x){=}f(x-h)+b ##

Coordinate system shifted by ## + (h,b)\, : \, f(x)=f(x+h)-b##
 
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Related to Correlation between shifting graph of a function and shifting the axes

1. How does shifting the graph of a function affect the overall appearance of the function?

Shifting the graph of a function changes its position on the coordinate plane, but does not alter the shape or slope of the function. It simply moves the entire graph up, down, left, or right.

2. Can shifting the axes change the behavior of a function?

No, shifting the axes does not change the behavior of a function. It only changes the reference point for the function's values on the coordinate plane.

3. What is the difference between shifting the graph and shifting the axes?

Shifting the graph involves moving the entire function up, down, left, or right. Shifting the axes involves changing the reference point for the function's values on the coordinate plane, without moving the function itself.

4. How do you determine the direction of the shift when shifting the graph or axes?

The direction of the shift is determined by the sign of the shift value. A positive shift value will move the graph or axes to the right or up, while a negative shift value will move them to the left or down.

5. Can a function be shifted both horizontally and vertically?

Yes, a function can be shifted both horizontally and vertically. This is known as a diagonal shift and involves changing the x and y coordinates of the function's points simultaneously.

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