Horizontal/vertical Stretch formatting help

[IDunno]

First time posting here, so I'm not sure if posting "homework" questions is allowed, but I'm going to ask anyway since I cant find any answers to my questions anywhere else;

Question: Describe the compression or expansion starting with the original relation

a) y=x^2 + 7 --> y=(1/4)x^2 + 7

b) y=x^2 + 2 --> y=4x^2 + 2

My answers:
a) vertical stretch BAFO 1/4

b) vertical stretch BAFO 2 (not sure how I got this one)

but the books answer was "Horizontal expansion BAFO 2" for "a" and "horizontal compression BAFO 1/2" for "b".

What I want to know is:

- even after realizing they were making horizontal transformations, shouldn't "a" be a Horizontal expansion BAFO 4 and "b" be a Horizontal expansion BAFO 1/4 since the stretches are BAFO (1/b)?

- shouldn't the question have been formatted as ((1/4)x)^2 + 7 and (4x)^2 + 2 if it was a horizontal transformation?

Thanks in advance for the help

 

[Dr.Peterson]

Question: Describe the compression or expansion starting with the original relation

a) y=x^2 + 7 --> y=(1/4)x^2 + 7

b) y=x^2 + 2 --> y=4x^2 + 2

My answers:
a) vertical stretch BAFO 1/4

b) vertical stretch BAFO 2 (not sure how I got this one)

but the books answer was "Horizontal expansion BAFO 2" for "a" and "horizontal compression BAFO 1/2" for "b".

I take it that BAFO means "by a factor of"; I've never seen that usage.

In (a), a vertical stretch by a factor of 1/4 would result in y = (1/4)(x^2 + 7) = (1/4)x^2 + 28. So that is wrong. That stretch would have to be combined with another transformation.

Similarly, in (b), a vertical stretch by a factor of 2 would result in y = 2(x^2 + 2) = 2x^2 + 4. You got that one totally wrong.

But in (a), a horizontal stretch by a factor of 2 replaces x with x/2, resulting in y = (x/2)^2 + 7 = (1/4)x^2 + 7, which is just what you want. And in (b), a horizontal compression by a factor of 1/2 replaces z with 2x, resulting in y = (2x)^2 + 2 = 4x^2 + 2.

- even after realizing they were making horizontal transformations, shouldn't "a" be a Horizontal expansion BAFO 4 and "b" be a Horizontal expansion BAFO 1/4 since the stretches are BAFO (1/b)?

Horizontal transformations are "inside" transformations, replacing x with something. In order to see them, you have to rewrite the functions in that form. That changes the 4's to 2's.

- shouldn't the question have been formatted as ((1/4)x)^2 + 7 and (4x)^2 + 2 if it was a horizontal transformation?

No, writing it in that form is your job! This is called "making you think" rather than spoon-feeding you a pre-digested problem.

And ((1/4)x)^2 + 7 is not the same as (1/4)x^2 + 7. You do see that, right?

Here is how I would approach (a):

We see that a vertical stretch would change the constant term (the y-intercept), so they must be looking for a horizontal transformation. Therefore I would "pull the factor inside" by rewriting (1/4)x^2 as (1/2)^2 x^2 = (x/2)^2.
Then, looking at y = (x/2)^2 + 7, I see that this is a horizontal stretch.

 

[IDunno]

I take it that BAFO means "by a factor of"; I've never seen that usage.

In (a), a vertical stretch by a factor of 1/4 would result in y = (1/4)(x^2 + 7) = (1/4)x^2 + 28. So that is wrong. That stretch would have to be combined with another transformation.

Similarly, in (b), a vertical stretch by a factor of 2 would result in y = 2(x^2 + 2) = 2x^2 + 4. You got that one totally wrong.

But in (a), a horizontal stretch by a factor of 2 replaces x with x/2, resulting in y = (x/2)^2 + 7 = (1/4)x^2 + 7, which is just what you want. And in (b), a horizontal compression by a factor of 1/2 replaces z with 2x, resulting in y = (2x)^2 + 2 = 4x^2 + 2.

Yeah BAFO is "by a factor of", sorry I should have been more specific.

Ok that makes sense, I was too focused on the number and completely forgot that it too was affected by the exponent. for some reason I also didn't think that the transformed relation was the end point, the question was asking me to essentially find the "middle" of the equation, not the end result (if that makes sense).

And yes, I know that ((1/4)x)^2 + 7 and (1/4)x^2 + 7 are not the same equation, that's what I was confused about. the equation format in the book using "a" as an example, at least in my eyes, could be either ((1/4)x)^2 + 7 or (1/4)x^2 + 7 as there are no brackets whatsoever in the original question.

 

[Dr.Peterson]

And yes, I know that ((1/4)x)^2 + 7 and (1/4)x^2 + 7 are not the same equation, that's what I was confused about. the equation format in the book using "a" as an example, at least in my eyes, could be either ((1/4)x)^2 + 7 or (1/4)x^2 + 7 as there are no brackets whatsoever in the original question.

Do you or do you not understand that without the brackets, it does NOT mean that the 1/4 is part of what is squared? What the problem says has to be read as written. You can't imagine additional parentheses that are not there, and which would change the meaning.

What is confusing you, as far as I can tell, is simply that they didn't make it easy for you by writing it in a form that made the required transformation immediately visible. It is common for students to get used to examples that are set up that way, and trip over problems that require an extra first step. And that is undoubtedly why this problem was written as it was: to give you practice in correcting wrong expectations.

 

[IDunno]

Do you or do you not understand that without the brackets, it does NOT mean that the 1/4 is part of what is squared? What the problem says has to be read as written. You can't imagine additional parentheses that are not there, and which would change the meaning.

What is confusing you, as far as I can tell, is simply that they didn't make it easy for you by writing it in a form that made the required transformation immediately visible. It is common for students to get used to examples that are set up that way, and trip over problems that require an extra first step. And that is undoubtedly why this problem was written as it was: to give you practice in correcting wrong expectations.

So what you're saying is that I'm focusing too much on the formatting of the question, and because of that, I'm getting confused and tripping up over the answer?

 

[Dr.Peterson]

So what you're saying is that I'm focusing too much on the formatting of the question, and because of that, I'm getting confused and tripping up over the answer?

Probably.

 

[IDunno]

Probably.

ok, thanks for the help!