VCE Methods Question Thread!

[keltingmeith]

Can someone please explain to me the transformations which lead you from what I circled in green to what I circled in red?
I.e. how to go from: \[(x+1)^2 -1\]
to \[16((x-\frac{1}{2})^2 -\frac{1}{16})\]
I put them both into "standard" form to make the transformations more clear. But I'm confused as to how my answer is incorrect.
My answer is:  dilate 16 units away from the x axis, followed by a translation of 3/2 to the right and 15/16 up.
The book's answer is:  dilate by a factor of 1/4 from the y-axis, then translation 3/4 units to the right
I considered using the matrix transformation thing but I'm not sure

This is from Cambridge Chapter 8, Q9b page 334

The thing with transformations questions, is there's always an infinite number of different transformations you can make to get from one equation to another. So, just because they have a different answer to you, doesn't mean that your answer is wrong. So, let's see what your transformation does.

Now, what I'm about to do might seem new to you - don't be scared by it, it's just yet another method to describing transformations. I personally think it's more logical than other ways of working through it, but you don't have to use it. It's only for working out equations, though.

So, you have x and y and they map to two new variables - x' and y'. So currently, we have:

x' -> x
y' -> y

Next, you dilate 16 units away from the x-axis. That is, you take every point on the graph, and take the y-coordinate, and multiply it by 16, giving us:

x' -> x
y' -> 16y

Next, you move 3/2 units to the right. So, you add 3/2 to x:

x' -> x + 3/2
y' -> 16y

Finally, you move 15/16 units up. This gives us:

x' -> x + 3/2
y' -> 16y + 15/16

Finally, to put this into the new equation, we need to solve them for x and y - so, let's do that:



And finally, put these into your original equation:



Hopefully I don't need to factorise this for you to see the only thing you did wrong was a translation in y. In fact, your answer is correct if you don't include the translation up by 15/16 units - give it a shot, see what happens.

[Mattjbr2]

The thing with transformations questions, is there's always an infinite number of different transformations you can make to get from one equation to another. So, just because they have a different answer to you, doesn't mean that your answer is wrong. So, let's see what your transformation does.

Now, what I'm about to do might seem new to you - don't be scared by it, it's just yet another method to describing transformations. I personally think it's more logical than other ways of working through it, but you don't have to use it. It's only for working out equations, though.

So, you have x and y and they map to two new variables - x' and y'. So currently, we have:

x' -> x
y' -> y

Next, you dilate 16 units away from the x-axis. That is, you take every point on the graph, and take the y-coordinate, and multiply it by 16, giving us:

x' -> x
y' -> 16y

Next, you move 3/2 units to the right. So, you add 3/2 to x:

x' -> x + 3/2
y' -> 16y

Finally, you move 15/16 units up. This gives us:

x' -> x + 3/2
y' -> 16y + 15/16

Finally, to put this into the new equation, we need to solve them for x and y - so, let's do that:



And finally, put these into your original equation:



Hopefully I don't need to factorise this for you to see the only thing you did wrong was a translation in y. In fact, your answer is correct if you don't include the translation up by 15/16 units - give it a shot, see what happens.

Amazing response! However, suppose we have no idea what transformations are needed, and are just given these two equations and are asked to provide the transformations needed to from one to the other. How would you go about doing that? I.e. how do you find the answer to this question and any other question like it?

[Lethal Sauce]

Amazing response! However, suppose we have no idea what transformations are needed, and are just given these two equations and are asked to provide the transformations needed to from one to the other. How would you go about doing that? I.e. how do you find the answer to this question and any other question like it?

I don't know if this will be much help to you, but I had a small section on this on my bound reference, which I have attached here. Hopefully it is clear enough to be of some help!

[Mattjbr2]

Can someone please tell me what's wrong with my logic here?
With 16 outside the brackets, there seems to be a y translation
When I move 16 into the brackets, the y translation disappears.?

Edit: I know that when I graph both functions there clearly is no y translation, however my y/y' method shows a y translation, does it not?

[Syndicate]

Can someone please tell me what's wrong with my logic here?
With 16 outside the brackets, there seems to be a y translation
When I move 16 into the brackets, the y translation disappears.?

Edit: I know that when I graph both functions there clearly is no y translation, however my y/y' method shows a y translation, does it not?

There is no specific answer for transformations. From what I see, your first method seems good

Example: y = 4x^2
So you can either have a dilation by a factor of 4 from the x-axis or dilation by a factor of 1/2 from the y-axis

[Mattjbr2]

There is no specific answer for transformations. From what I see, your first method seems good

Example: y = 4x^2
So you can either have a dilation by a factor of 4 from the x-axis or dilation by a factor of 1/2 from the y-axis

What about the y translation? There isn't a y translation between the graphs, but the 1st photo (with a red X) shows a y translation. Doesn't it?

Edit: attached the photo here. Doesn't it show a y+15 translation between the graphs in line 5?

[keltingmeith]

What about the y translation? There isn't a y translation between the graphs, but the 1st photo (with a red X) shows a y translation. Doesn't it?

Edit: attached the photo here. Doesn't it show a y+15 translation between the graphs in line 5?

It's because when you do the dilation from the X-axis, you throw out the current positioning of the graph, so you need to re-translate it. This isn't an issue of you do the dilation from the y-axis.

[VanillaRice]

What about the y translation? There isn't a y translation between the graphs, but the 1st photo (with a red X) shows a y translation. Doesn't it?

Edit: attached the photo here. Doesn't it show a y+15 translation between the graphs in line 5?

Remembering that there is no specific order for transformations, have a think about how a vertical translation can be involved in a set of transformations, but not appear in the end product.

Let's take your example

And we return to a function where there is no "apparent" vertical translation.

Hope that helps

[Mattjbr2]

It's because when you do the dilation from the X-axis, you throw out the current positioning of the graph, so you need to re-translate it. This isn't an issue of you do the dilation from the y-axis.

Cheers!

Remembering that there is no specific order for transformations, have a think about how a vertical translation can be involved in a set of transformations, but not appear in the end product.

Let's take your example

And we return to a function where there is no "apparent" vertical translation.

Hope that helps

Perfect! Thank you!

[snowisawesome]

What's the best way to prepare for a year 12 sac and exam in methods?
Any efficient preparation tips to get great results would be greatly appreciated
How achievable is a 35 raw?

[zhen]

What's the best way to prepare for a year 12 sac and exam in methods?
Any efficient preparation tips to get great results would be greatly appreciated
How achievable is a 35 raw?

For your year 12 SACs you should really get on top of your coursework and do all the exercises from the textbook. Also, doing some exam style practice questions from checkpoints or practice exams can really help to prepare for SACs and lets you get used to these questions before the exam. Moreover, sometimes teachers give you practice SACs to do. Make sure you do these before the SACs. For the exam, personally doing practice exams was the best way I prepared for the exam. Make sure you do these in timed conditions so that you get used to doing things with time pressure. Also, closer to the exam I’d start to pick out the hardest questions and only do them. After 10 or so practice exams you’ll notice a pattern and you’ll notice that there are only certain parts of the exam you find difficult. So, doing hard multiple choice or extended response questions really helped me to get better at doing these hard questions. Anyway, I don’t know how proficient you are at maths, but if you try your hardest then 35 should be achievable. 

[snowisawesome]

Thanks Zhen!
Also, did you mean that we should be doing every single textbook question, or just the ones our teachers set us?
Thanks

[zhen]

Thanks Zhen!
Also, did you mean that we should be doing every single textbook question, or just the ones our teachers set us?
Thanks

I only did the ones the teachers set me, but I did a lot of additional questions such as ones from old practice exams and other sources. I thought doing these questions were much more helpful than some of the textbook questions that get really repetitive. Just make sure to always do the extended response questions at the end of each exercise, which are quite challenging so you can learn a lot for them.

[snowisawesome]

Is the extended response always tech-active?

[zhen]

Is the extended response always tech-active?

In VCAA exams extended response is tech active I think. However, some schools give hard extended response questions for their tech free SACs. I personally did as much as I could by hand, which sometimes is the entire question and I checked using my CAS or did the final calculations using my CAS.