Transformation in logarithm

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KingRich
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#1
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#1
Can someone confirm if this transformation is possible using geometric transformation only?


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From first observation
y=In(x)———-> y=In(x+1) in part of the final transformation, informs me that there must be a translation one unit left.

Mmm, i don’t believe this possible by a geometrical stretch, although I could be wrong.

Confirmation of the question being possible would be sufficient as I’ll have to rethink it.

Thank you in advance
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mqb2766
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(Original post by KingRich)
Can someone confirm if this transformation is possible using geometric transformation only?


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From first observation
y=In(x)———-> y=In(x+1) in part of the final transformation, informs me that there must be a translation one unit left.

Mmm, i don’t believe this possible by a geometrical stretch, although I could be wrong.

Confirmation of the question being possible would be sufficient as I’ll have to rethink it.

Thank you in advance
Sounds like you have the right sort of idea. Are you thinking about a horizontal or vertical stretch?
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KingRich
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(Original post by mqb2766)
Sounds like you have the right sort of idea. Are you thinking about a horizontal or vertical stretch?
I assume by geometric transformation, it means that it has to be achieved by means of horizontal and vertical stretch only, which would refer to a geometric transformation, example: y=p(fx) or y=f(2x) as a translation would mean adding or subtracting which is an arithmetic movement ( y=f(x+a) or y=f(x)+c. I know that you know the formulas. I’m just stating to show my observations lol

I could be wrong in that thinking
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mqb2766
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(Original post by KingRich)
I assume by geometric transformation, it means that it has to be achieved by means of horizontal and vertical stretch only, which would refer to a geometric transformation, example: y=p(fx) or y=f(2x) as a translation would mean adding or subtracting which is an arithmetic movement ( y=f(x+a) or y=f(x)+c. I know that you know the formulas. I’m just stating to show my observations lol

I could be wrong in that thinking
A geometric transformation is a shift and/or scale in either (or both) the x and y directions.

So if
f(x) = ln(x)
can you write down the (3) transformations? Youve got the first one (a horizontal translation) and you seem to "know" the other two. Note that when you have a nonlinear function like ln(x), you apply shift and scale on x "inside" the ln() and shift and scale on y to "outside" of ln(). It may help to write
y+2 = 4 ln(x+1)

Note your graph is about right, but the transformed function would be ~4 times steeper than the original one (when x ~ x+1). Your sketch looks like a vertical shift up rather than being steeper. A quick plot in desmos would illustrate things.
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KingRich
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(Original post by mqb2766)
A geometric transformation is a shift and/or scale in either (or both) the x and y directions.

So if
f(x) = ln(x)
can you write down the (3) transformations? Youve got the first one (a horizontal translation) and you seem to "know" the other two. Note that when you have a nonlinear function like ln(x), you apply shift and scale on x "inside" the ln() and shift and scale on y to "outside" of ln(). It may help to write
y+2 = 4 ln(x+1)

Note your graph is about right, but the transformed function would be ~4 times steeper than the original one (when x ~ x+1). Your sketch looks like a vertical shift up rather than being steeper. A quick plot in desmos would illustrate things.
This is my initial assessment when I first observe.

Horizontal translation left by one unit, followed by a vertical translation down by two units, then a vertical stretch, by factor 4.

y=In(x) —-> y=In(x+1) ——>y=In(x+1)-2 ——-> y=4In(x+1)-2….

so, a geometrical transformation is both translation and stretch?
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mqb2766
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(Original post by KingRich)
This is my initial assessment when I first observe.

Horizontal translation left by one unit, followed by a vertical translation down by two units, then a vertical stretch, by factor 4.

y=In(x) —-> y=In(x+1) ——>y=In(x+1)-2 ——-> y=4In(x+1)-2….

so, a geometrical transformation is both translation and stretch?
https://courses.lumenlearning.com/bo...ansformations/
Could also include rotations and reflections.

Not quite.
y=In(x) —-> y=In(x+1) ——>y=In(x+1)-2 ——-> y=4(In(x+1)-2) = 4ln(x+1) - 8
If you scale after the y shift, the shift gets affected. So either y shift after the y scale or shift by a different value so that after the scaling its ok.
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KingRich
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(Original post by mqb2766)
https://courses.lumenlearning.com/bo...ansformations/
Could also include rotations and reflections.

Not quite.
y=In(x) —-> y=In(x+1) ——>y=In(x+1)-2 ——-> y=4(In(x+1)-2) = 4ln(x+1) - 8
If you scale after the y shift, the shift gets affected. So either y shift after the y scale or shift by a different value so that after the scaling its ok.
Argh!! Yes. So, the stretch before the translation. I know what I meant lol. I’ve not learnt rotations as of yet, is this beyond A-Level?
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mqb2766
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(Original post by KingRich)
Argh!! Yes. So, the stretch before the translation. I know what I meant lol. I’ve not learnt rotations as of yet, is this beyond A-Level?
I think its further maths (matrices).

Thats why its good to write it as
y+2 = 4 ln(x+1)
You could also do a y shift by -1/2 then scale by 4, and that would give the same result. But doing the scale, then shift is the easiest thing to do. Its a common mistake to make so try not to.
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KingRich
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(Original post by mqb2766)
I think its further maths (matrices).

Thats why its good to write it as
y+2 = 4 ln(x+1)
You could also do a y shift by -1/2 then scale by 4, and that would give the same result. But doing the scale, then shift is the easiest thing to do. Its a common mistake to make so try not to.
I shall give the rotations a miss for now then, unless I consider pursuing further but I highly doubt it lol

oh, that’s true. Thanks for the advice. Always appreciated
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