Vertical and Horizontal Stretch vs. Compression confusion

[barcacookie]

If you know what f(x) is and g(x) = 1/2f[2(x-1)]+4

There is a vertical stretch by a factor of 1/2, and a horizontal stretch by a factor of 1/2 because you would have to multiply all previous input values by 1/2 to get the same output as f(x). What would the vertical and horizontal compression be and why? Would the horizontal compression be by a factor of 2 because you would have to divide all previous input values by 2 to get the same output? But this logic also contrasts what I saw online in which said that it is a hori. compression by a factor of 1/k when k>1. In this example, k=2, so if I followed what I saw online then this would be a horizontal compression by a factor of 1/2? I am also confused because when I search online, sources tell me that when a > 1, there is a vertical stretch by a factor of "a", but in my case a < 1 and I believe it is still a vert. stretch by a factor of 1/2. Would this also be a vertical compression by a factor of 1/2 based on what I saw online?

I am very confused because what I read online contrasts what I have been taught, as I believe that you could call most transformations affecting "a" and "k" a stretch but maybe not a compression.

Also, on one question I was told to map f(x) = sqrt(x) after I apply multiple transformations. Of course, to do this I would have to take these transformations and create a mapping rule.

One of the transformations was hori. compression by a factor of 3.

The answer for the mapping rule if this was the only transformation would be (1/3x, y).

I am confused about why it would be 1/3x because then that means that k = 3. If what I read online was true then this would be a hori. compression by a factor of 1/k, which would be 1/3. but the transformation in the question was clearly stated as a hori. compression by a factor of 3, which would have to mean that k = 1/3 because 1/ 1/3 = 3.

Please help clarify for me my errors.

 

[Dr.Peterson]

If you know what f(x) is and g(x) = 1/2f[2(x-1)]+4

There is a vertical stretch by a factor of 1/2, and a horizontal stretch by a factor of 1/2 because you would have to multiply all previous input values by 1/2 to get the same output as f(x). What would the vertical and horizontal compression be and why? Would the horizontal compression be by a factor of 2 because you would have to divide all previous input values by 2 to get the same output? But this logic also contrasts what I saw online in which said that it is a hori. compression by a factor of 1/k when k>1. In this example, k=2, so if I followed what I saw online then this would be a horizontal compression by a factor of 1/2? I am also confused because when I search online, sources tell me that when a > 1, there is a vertical stretch by a factor of "a", but in my case a < 1 and I believe it is still a vert. stretch by a factor of 1/2. Would this also be a vertical compression by a factor of 1/2 based on what I saw online?

I am very confused because what I read online contrasts what I have been taught, as I believe that you could call most transformations affecting "a" and "k" a stretch but maybe not a compression.

Also, on one question I was told to map f(x) = sqrt(x) after I apply multiple transformations. Of course, to do this I would have to take these transformations and create a mapping rule.

One of the transformations was hori. compression by a factor of 3.

The answer for the mapping rule if this was the only transformation would be (1/3x, y).

I am confused about why it would be 1/3x because then that means that k = 3. If what I read online was true then this would be a hori. compression by a factor of 1/k, which would be 1/3. but the transformation in the question was clearly stated as a hori. compression by a factor of 3, which would have to mean that k = 1/3 because 1/ 1/3 = 3.

Please help clarify for me my errors.

This is an area where it is dangerous to get information online, because different sources use different terminology, which sometimes seems (or actually is) contradictory.

You should use the terminology of your own classroom and textbook, and not worry if others differ.

In particular, if (as your initial example suggests) you were taught to use the term "stretch" for both increasing and decreasing the height or width, then you don't need to use the term "compress". For you, "stretching" by a factor of 1/2 is the same thing that others call either compressing by a factor of 1/2, or compressing by a factor of 2! So you can just go by the factor: if it is less than 1/2, then you are really compressing, though you call it a stretch. (This is not illogical, though it is contrary to everyday usage: it is similar to saying the temperature "increased by -5 degrees", meaning that it actually decreased 5 degrees.)

But your second example indicates that your textbook does use the word "compress". You will have to find out how it uses the factor in such a case. This is where sources differ considerably. From your stated answer, it appears that they take "compress by a factor greater than 1" to be an actual compression (making it smaller).

Here is a place where I surveyed a number of sources to show the variety of usages. Your own appears to be form Ia or Ib, at least when using the word "compress". Can you quote for me what they say when they first introduce what a stretch or compression means, showing their terminology?

It is unfortunate that things are so inconsistent; I don't know how this happened, but I suspect it's because educators, rather than mathematicians, coined the various usages. Educators talk to their own students; mathematicians talk to one another, and need a common language.

 

[barcacookie]

This is an area where it is dangerous to get information online, because different sources use different terminology, which sometimes seems (or actually is) contradictory.

You should use the terminology of your own classroom and textbook, and not worry if others differ.

In particular, if (as your initial example suggests) you were taught to use the term "stretch" for both increasing and decreasing the height or width, then you don't need to use the term "compress". For you, "stretching" by a factor of 1/2 is the same thing that others call either compressing by a factor of 1/2, or compressing by a factor of 2! So you can just go by the factor: if it is less than 1/2, then you are really compressing, though you call it a stretch. (This is not illogical, though it is contrary to everyday usage: it is similar to saying the temperature "increased by -5 degrees", meaning that it actually decreased 5 degrees.)

But your second example indicates that your textbook does use the word "compress". You will have to find out how it uses the factor in such a case. This is where sources differ considerably. From your stated answer, it appears that they take "compress by a factor greater than 1" to be an actual compression (making it smaller).

Here is a place where I surveyed a number of sources to show the variety of usages. Your own appears to be form Ia or Ib, at least when using the word "compress". Can you quote for me what they say when they first introduce what a stretch or compression means, showing their terminology?

It is unfortunate that things are so inconsistent; I don't know how this happened, but I suspect it's because educators, rather than mathematicians, coined the various usages. Educators talk to their own students; mathematicians talk to one another, and need a common language.

My teacher told me that the key difference between a stretch and a compression was that a stretch is a multiplication and a compression is a division. From my notes, it says that a vert. stretch is a transformation that results in the distance from the x-axis every point growing by a scale factor greater than 1 and a vertical compression is a transformation that results in the distance from the x-axis of every point shrinking by a scale factor between 0 and 1. I then checked my textbook and the definition for stretch and compression are the same but it added on that a horizontal stretch is the distance from the y-axis of every point growing by a scale factor greater than 1, while a horizontal compression is the distance from the y-axis of every point shrinking by a scale factor between 0 and 1.

It also states that:

If a> 1, the graph is a vert. stretch by a factor of a.
If 0<a<1, the graph is a vert compression by a factor of a.

If k>1, the graph is a horizontal compression by a factor of 1/k
If 0<k<1, the graph is a hori. strech by a factor of 1/k.

But my teacher also stated that the textbook was wrong, which is shown as I can say a vert./hori. stretch for most transformations.

Btw, the answers to the questions I posted above were by my teacher, not the textbook.

 

[Dr.Peterson]

My teacher told me that the key difference between a stretch and a compression was that a stretch is a multiplication and a compression is a division. From my notes, it says that a vert. stretch is a transformation that results in the distance from the x-axis every point growing by a scale factor greater than 1 and a vertical compression is a transformation that results in the distance from the x-axis of every point shrinking by a scale factor between 0 and 1. I then checked my textbook and the definition for stretch and compression are the same but it added on that a horizontal stretch is the distance from the y-axis of every point growing by a scale factor greater than 1, while a horizontal compression is the distance from the y-axis of every point shrinking by a scale factor between 0 and 1.

It also states that:

If a> 1, the graph is a vert. stretch by a factor of a.
If 0<a<1, the graph is a vert compression by a factor of a.

If k>1, the graph is a horizontal compression by a factor of 1/k
If 0<k<1, the graph is a hori. strech by a factor of 1/k.

But my teacher also stated that the textbook was wrong, which is shown as I can say a vert./hori. stretch for most transformations.

Btw, the answers to the questions I posted above were by my teacher, not the textbook.

Okay, good. What we see first is that your teacher and textbook both use scale factor (as I would) to mean the amount by which a coordinate is multiplied: greater than 1 = stretch, less than 1 = compress/shrink.

On the other hand, you were told that compression means division; there you are thinking of division by a number greater than 1, which is equivalent to multiplying by a number less than 1. This is not contradictory, just a reminder to be careful!

I don't think you've ever stated what you mean by a and k; evidently they refer to the form

g(x) = af(kx)

or a more complete form like

g(x) = af(kx + c) + b

What the book says about them agrees with the terminology I called IIa, where the factor is always the multiplier. I don't see any contradictions between the book and teacher, except perhaps that your teacher also allows you to say what you did initially, describing a compression alternatively as "a vertical stretch by a factor of 1/2". That's not really contradictory, but supplementary.

Are we agreed on this?

Now let's look back at your specific question:

Also, on one question I was told to map f(x) = sqrt(x) after I apply multiple transformations. Of course, to do this I would have to take these transformations and create a mapping rule.

One of the transformations was hori. compression by a factor of 3.

The answer for the mapping rule if this was the only transformation would be (1/3x, y).

I am confused about why it would be 1/3x because then that means that k = 3. If what I read online was true then this would be a hori. compression by a factor of 1/k, which would be 1/3. but the transformation in the question was clearly stated as a hori. compression by a factor of 3, which would have to mean that k = 1/3 because 1/ 1/3 = 3.

Based on what you've said, you are always using "scale factor" to mean a multiplier; so in your class's usage, you can't say "compression by a factor of 3", though others do. You would say compression by a factor of 1/3 (that is, multiplying by 1/3). So the phrasing of the problem is incompatible with what you have told me about both your teacher and your textbook! But even so, because it is described as a compression, you can tell that the answer has to be what you state, (1/3 x, y).

The important thing, I think, is not to blindly follow rules that whatever factor you are told is the reciprocal of k. Since this problem is stated with a "scale factor" greater than 1 for a compression, you know that the actual scale factor as you are using it in your class must be 1/3 rather than 3, so your k is 3.

Does this clear things up?

 

[barcacookie]

Okay, good. What we see first is that your teacher and textbook both use scale factor (as I would) to mean the amount by which a coordinate is multiplied: greater than 1 = stretch, less than 1 = compress/shrink.

On the other hand, you were told that compression means division; there you are thinking of division by a number greater than 1, which is equivalent to multiplying by a number less than 1. This is not contradictory, just a reminder to be careful!

I don't think you've ever stated what you mean by a and k; evidently they refer to the form

g(x) = af(kx)

or a more complete form like

g(x) = af(kx + c) + b

What the book says about them agrees with the terminology I called IIa, where the factor is always the multiplier. I don't see any contradictions between the book and teacher, except perhaps that your teacher also allows you to say what you did initially, describing a compression alternatively as "a vertical stretch by a factor of 1/2". That's not really contradictory, but supplementary.

Are we agreed on this?

Now let's look back at your specific question:



Based on what you've said, you are always using "scale factor" to mean a multiplier; so in your class's usage, you can't say "compression by a factor of 3", though others do. You would say compression by a factor of 1/3 (that is, multiplying by 1/3). So the phrasing of the problem is incompatible with what you have told me about both your teacher and your textbook! But even so, because it is described as a compression, you can tell that the answer has to be what you state, (1/3 x, y).

The important thing, I think, is not to blindly follow rules that whatever factor you are told is the reciprocal of k. Since this problem is stated with a "scale factor" greater than 1 for a compression, you know that the actual scale factor as you are using it in your class must be 1/3 rather than 3, so your k is 3.

Does this clear things up?

But a hori. compression by a factor of 3 is in the question, yet I cannot say it?
Is it because "k" has to be greater than 1, so when writing "horizontal compression by a factor of 1/k", 1/k would never be greater than 1 since k has to be greater than 1?
I am also confused as to what you mean "not to blindly follow rules that whatever factor you are told is the reciprocal of k" because "horizontal compression/stretch by a factor of 1/k" is part of my textbook.

 

[Dr.Peterson]

But a hori. compression by a factor of 3 is in the question, yet I cannot say it?
Is it because "k" has to be greater than 1, so when writing "horizontal compression by a factor of 1/k", 1/k would never be greater than 1 since k has to be greater than 1?
I am also confused as to what you mean "not to blindly follow rules that whatever factor you are told is the reciprocal of k" because "horizontal compression/stretch by a factor of 1/k" is part of my textbook.

First, please clarify: Is the question from your textbook, your teacher, or another source? It seems to be worded contrary to what they both teach. You can say what you want, but it seems to me that your text and teacher would not ask the question this way.

What I mean by not blindly following rules (which is a general principle in learning any math well) is that you must think about what a question means. In this case, as I pointed out, different sources word these questions differently, and not everyone uses "k" in the same way. Some describe the compression as "by a factor of 3", but others as "by a factor of 1/3" as in the examples you gave. So the fact that the number 3 is used in the question does not necessarily mean that is the value of 1/k. You have to determine from context whether it is 3 or 1/3.

In the way you are using k (the coefficient of x), it is not, in general, restricted to be greater than 1. It is true, however, that in a compression, 1/k will always be less than 1, so k will be greater than 1. Note that this is the reverse of what you said, in terms of causality: BECAUSE compression must involve multiplying the x-coordinate by a number smaller than 1, THEREFORE 1/k must be smaller than 1, so k must be greater than 1.

 

[barcacookie]

First, please clarify: Is the question from your textbook, your teacher, or another source? It seems to be worded contrary to what they both teach. You can say what you want, but it seems to me that your text and teacher would not ask the question this way.

What I mean by not blindly following rules (which is a general principle in learning any math well) is that you must think about what a question means. In this case, as I pointed out, different sources word these questions differently, and not everyone uses "k" in the same way. Some describe the compression as "by a factor of 3", but others as "by a factor of 1/3" as in the examples you gave. So the fact that the number 3 is used in the question does not necessarily mean that is the value of 1/k. You have to determine from context whether it is 3 or 1/3.

In the way you are using k (the coefficient of x), it is not, in general, restricted to be greater than 1. It is true, however, that in a compression, 1/k will always be less than 1, so k will be greater than 1. Note that this is the reverse of what you said, in terms of causality: BECAUSE compression must involve multiplying the x-coordinate by a number smaller than 1, THEREFORE 1/k must be smaller than 1, so k must be greater than 1.

This question was from my teacher. I really appreciate the help you have been given me. Thank you. As well, to clarify in my first question, "If you know what f(x) is and g(x) = 1/2f[2(x-1)]+4", then as well to being a vertical stretch by a factor of 1/2, this would also be called a vertical compression by a factor of 1/2 since 0<a<1?

 

[Dr.Peterson]

This question was from my teacher. I really appreciate the help you have been given me. Thank you. As well, to clarify in my first question, "If you know what f(x) is and g(x) = 1/2f[2(x-1)]+4", then as well to being a vertical stretch by a factor of 1/2, this would also be called a vertical compression by a factor of 1/2 since 0<a<1?

Right, both ways of saying it make sense, since the 1/2 implies compression regardless.

I assume the question you said is from your teacher is the second, "One of the transformations was hori. compression by a factor of 3", and you're okay with it now. I'm still curious whether the teacher is happy with anyone using either 3 or 1/3 there.