Transformations of Graphs

[Monkeyseat]

y = x^2
y = x^3
y = 1/x
y = cosx
y = sinx

Describe the transformation to the following functions from the ones above:

1) y = 2(x + 1)^3

x^3 graph translated by vector [-1, 0] and stretched by a scale factor of 2 in the y axis.

2) y = 3 - x^2

x^2 reflected in the x axis, translated by vector [0, 3].

3) y = 2sin3x

sinx stretched by a scale factor of 2 in the y axis, and stretched by a scale factor 1/3 in the x axis.

--

Ok, so that's what I thought when doing them but can they all be done in one single transformation? I presumed so, mainly because it says transformation (singular) and all the other ones I did as single transformations.

Can anyone give me some suggestions?

Many thanks.

By the way, I don't know how to type vectors on the computer, so apologies about that.

 

[wjm11]

Monkeyseat,

Your interpretations are correct, though in problem 2), I would not use the term "scale factor" to describe the translation: ("translated by a scale factor of [0, 3]"). I believe the use of the term "transformation" is intended to include all translations, scalings, and reflections that have been applied.

 

[Monkeyseat]

Monkeyseat,

Your interpretations are correct, though in problem 2), I would not use the term "scale factor" to describe the translation: ("translated by a scale factor of [0, 3]"). I believe the use of the term "transformation" is intended to include all translations, scalings, and reflections that have been applied.

Thanks for the reply. Sorry, I meant translation by vector not scale factor, that was just a mistake.

So these 3 questions cannot be translated in any other way i.e. in one transformation like by a single translation or a single scale factor? Can part (c) be transformed by one single scale factor instead of 2 "stretches"? It's just all the other questions we've been doing have been doable in one transformation, for example y = 5 + x^4 to y = 15 + 3x^4 is by a scale factor of 3 in the y direction. Just want to check.

Thanks!

 

[wjm11]

y = 5 + x^4 to y = 15 + 3x^4 is by a scale factor of 3 in the y direction

This statement is only true if you designate y = 5 + x^4 as your “original” or “parent” function. The way we describe a transformation is always in relation to the “parent” function we have defined.

If, on the other hand, you used y = x^4 as your “parent” function, then you would have to describe the transformation to y = 15 + 3x^4 as both a translation and a scaling.

You have all the “parent” functions defined at the top of your problem,

y = x^2
y = x^3
y = 1/x
y = cosx
y = sinx

so we must refer to them to decide what transformation has taken place.

Make sense?

 

[Monkeyseat]

y = 5 + x^4 to y = 15 + 3x^4 is by a scale factor of 3 in the y direction

This statement is only true if you designate y = 5 + x^4 as your “original” or “parent” function. The way we describe a transformation is always in relation to the “parent” function we have defined.

If, on the other hand, you used y = x^4 as your “parent” function, then you would have to describe the transformation to y = 15 + 3x^4 as both a translation and a scaling.

You have all the “parent” functions defined at the top of your problem,

y = x^2
y = x^3
y = 1/x
y = cosx
y = sinx

so we must refer to them to decide what transformation has taken place.

Make sense?

Yes I meant y = 5 + x^4 as the "parent function".

I know what you are saying, so in the questions I posted, the bottom line is we have to do more than one transformation out of a translation, reflection and scaling? I just want to make sure that the way I did it in my original post is the correct way before I write it down.

Thanks.

EDIT:

Can I just ask, in mapping y = x^2 onto y = 3 - x^2, before translating is it reflected in the x axis or y axis (although I presume you can do it either way around)?

I believed it was in the x axis but when I compared y = x^2 to y = -x^2 in on this website, it showed it to be a reflection in the y axis. I always thought y = -f(x) was reflected in the x axis and y = f(-x) was in the y axis, so y = -x^2 would be in the x axis and y = (-x)^2 would be in the y axis. Although on that website it shows y = x^2 to y = 3 - x^2 to be a translation and reflection in the x axis, I don't know why it differs...

Also in my textbook it says the function that maps y = 1 + x^3 onto y = 1 - x^3 is a reflection in the y axis, not the x axis like I put for part (2). This is similar to my problem number (2) so I thought it would be reflected in the x axis. Why is this different and reflected in the y axis? Or is mine wrong?

So I was wondering is y = x^2 to y = -x^2 reflected in the x or y axis? Is y = x^2 to y = 3 - x^2 reflected in the x or y axis?

Thanks for your time.

Much appreciated.

 

[wjm11]

I always thought y = -f(x) was reflected in the x axis and y = f(-x) was in the y axis, so y = -x^2 would be in the x axis and y = (-x)^2 would be in the y axis.

You are correct.

The website you referred to is very misleading. It seems to be treating –x^2 as (-x)^2. Try entering –(x^2) in this website, and it will give you the right answer.

Your graph in 2) is also correct.

You asked: So I was wondering is y = x^2 to y = -x^2 reflected in the x or y axis?

It is a reflection across the x axis. If you have a graphing calculator, try entering the equations into it and inspect the graphs. Your calculator will, hopefully, give you a different result than the above mentioned website.

 

[Monkeyseat]

I always thought y = -f(x) was reflected in the x axis and y = f(-x) was in the y axis, so y = -x^2 would be in the x axis and y = (-x)^2 would be in the y axis.

You are correct.

The website you referred to is very misleading. It seems to be treating –x^2 as (-x)^2. Try entering –(x^2) in this website, and it will give you the right answer.

Your graph in 2) is also correct.

You asked: So I was wondering is y = x^2 to y = -x^2 reflected in the x or y axis?

It is a reflection across the x axis. If you have a graphing calculator, try entering the equations into it and inspect the graphs. Your calculator will, hopefully, give you a different result than the above mentioned website.

Thanks for the reply, that website does seem to be a bit misleading. Unfortunately I don't have a graphing calculator at the moment, but I've found another online tool that plots them more accurately.

Can I just ask, in my textbook it says the function that maps y = 1 + x^3 onto y = 1 - x^3 is a reflection in the y axis, not the x axis like I put for part (2). This is similar to my problem number (2) so I thought it would be reflected in the x axis. Why is this different and reflected in the y axis?

So all 3 of these questions cannot be done in one single transformation?

Sorry to repeat stuff, just trying to clear this up.

Thanks.

 

[wjm11]

in my textbook it says the function that maps y = 1 + x^3 onto y = 1 - x^3 is a reflection in the y axis, not the x axis like I put for part (2). This is similar to my problem number (2) so I thought it would be reflected in the x axis. Why is this different and reflected in the y axis?

Please make an xy table and plot enough points to see what these functions look like. You will find that, yes, they are mirror images of each other across the y-axis. According to your rules for a y reflection, we would change f(x) into f(-x), right? Let see:

f(x) = 1 + x^3 so,
f(-x) = 1 + (-x)^3 = 1 – x^3

So, yes, this is a reflection across the y axis.

Note: We’re able to remove the () around –x in this case because the exponent is odd.

Once again, it is important to recognize what the parent function is. In the case you cite, the parent function is not y = x^3. If we were changing y = x^3 into y = 1-x^3, we would describe it as a reflection across the x-axis because of the (-), followed by a translation upward of 1.

One interesting thing to note in the case of y = x^3 is that the function is point symmetric about the origin (as opposed to axisymmetric about either the x or y axis). As a result,

-f(x) = f(-x)

for this function, meaning that a y reflection looks the same as an x reflection.

 

[Monkeyseat]

in my textbook it says the function that maps y = 1 + x^3 onto y = 1 - x^3 is a reflection in the y axis, not the x axis like I put for part (2). This is similar to my problem number (2) so I thought it would be reflected in the x axis. Why is this different and reflected in the y axis?

Please make an xy table and plot enough points to see what these functions look like. You will find that, yes, they are mirror images of each other across the y-axis. According to your rules for a y reflection, we would change f(x) into f(-x), right? Let see:

f(x) = 1 + x^3 so,
f(-x) = 1 + (-x)^3 = 1 – x^3

So, yes, this is a reflection across the y axis.

Note: We’re able to remove the () around –x in this case because the exponent is odd.

Once again, it is important to recognize what the parent function is. In the case you cite, the parent function is not y = x^3. If we were changing y = x^3 into y = 1-x^3, we would describe it as a reflection across the x-axis because of the (-), followed by a translation upward of 1.

One interesting thing to note in the case of y = x^3 is that the function is point symmetric about the origin (as opposed to axisymmetric about either the x or y axis). As a result,

-f(x) = f(-x)

for this function, meaning that a y reflection looks the same as an x reflection.

Sorry for taking a while to get back to you, but many thanks for helping.

This has been very useful.