Spesh 1/2 Question, please help!!!

[geminii]

Hi everyone!! So this is the question:

Question 17
The translation that maps the graph of y=(x-4)2 +5 to the graph of y=x2 can be described as:
A 4 units in the negative direction of the x axis and 5 units in the positive direction of the y axis
B 4 units in the positive direction of the x axis and 5 units in the positive direction of the y axis
C 4 units in the negative direction of the x axis and 5 units in the negative direction of the y axis
D 4 units in the positive direction of the x axis and 5 units in the negative direction of the y axis
E 5 units in the negative direction of the x axis and 4 units in the positive direction of the y axis

It would be great if someone could take me through how I solve this! I will probably post more questions soon. Thanks!!

[geminii]

Here are some others that I'm having trouble with:

Question 18
The equation of the image of the graph of y = x2 after a reflection in the y axis followed by a translation of 3 units in the positive direction of the x axis and 4 units in the negative direction of the y axis is:
A y=(-x-3)2 –4
B y=–(x-3)2 –4
C y=(x+3)2 –4
D y=(3-x)2 +4
E y=(3x-)2 -4

Question 19
A transformation has rule (x, y) -> (2x + 1, y – 3). The point (a, b) is transformed to the point (11, 9). The values of a and b are:
A a=5andb=12
B a=23andb=6
C a=8andb=5
D a=6andb=23
E a=3andb=–3

Question 20
The graph of the relation y = x2 is dilated by a factor of 2 from the y axis then translated so that (x, y) -> (x – 3, y + 2). The equation of the image is:
A y=41(x+3)2 +2
B y=4(x+3)2 –2
C y=41(x-3)2 +2
D y=4(x-3)2 +2
E y=41(x-3)2 -2

Thanks again!! 

[MightyBeh]

Hey Ace, here goes nothing? 
Assuming these are non-calculator, correct me if I'm wrong (both with my answers, and the calc/notes thing)

Question 17

So your graph is in the turning point form of a parabola, which thankfully is pretty easy to work with.
Here's the general form:

And here's yours:

So what does this actually mean? In the general form, 'a' represents the dilation (which we don't need to worry about, it's equal to 1 here), 'h' represents the displacement in the x-axis, and k represents the displacement and in y-axis.

At a glance we can say that the graph has been translated 5 positive units in the y-axis, because there's nothing messing with the k value. So we have already narrowed the answers down to A, B or E.

the horizontal translation is a little trickier (although it's not hard, don't panic! ) - the brackets are buttheads and they flip things around. A negative h value will actually give you a translation in the positive direction of the x-axis. That wouldleave us with B as our answer, but as Callum pointed out to me it's the translation from to , not the other way around (whoops ). For this problem, all that means is that we swap the positive/negative h and k values, so the answer is C, "4 units in the negative direction of the x axis and 5 units in the negative direction of the y axis"

Thanks for pointing that one out Callum! <3

It's a little tricky to explain with text, so check these out:
Video on transformations(youtube)
And this screenshot from my notes might be helpful too:

Question 18

Okay, so we know that the following has happened to our graph:

• "Reflection in the y-axis"
• "Translation of 3 units in the positive direction of the x-axis
• "Translation of 4 units in the negative direction of the y-axis"

So we just have to build an equation based off of that information and the turning point form of a parabola.



To go positive in the x-axis, we say that the value h is negative. Because we're moving three units, our h is 3. In our equation:

But wait; there hasn't been any dilation so we can knock out that pesky 'a' value:


Next, we have to move 4 units in the negative direction of the y-axis. This one's easy; there's no strings attached to k so we can just throw in our -4:


Now all we need to do is add in our reflection in the y-axis. The easiest way to describe this is we're making all our positive x-values negative, and all our negative x-values positive - or, we're multiplying x we're multiplying everything in the brackets by -1 to send it through the mathsy wardrobe  to some mathematical equivalent of Narnia. Well, except it's not a wardrobe it's the y-axis. 

So our final equation is:


Still not 100%? Check this out:
Mathwarehouse has a pretty neat explanation (with pictures!) and some practice problems.

Question 19

This one's just some algebra, thankfully. We have to find what a and b are. To do that we turn our compositions (not 100% on the name, the thing that shows you the change in the x and y values) into equations, like so:



Then we solve these for x (a) and y (b).


Just doing that narrows our selection of answers down to only one (A), but it's such a simple equation to solve that it's probably good practice to find b to make sure.



So (a,b) is (5,12)

Question 20

The graph of the relation y = x2 is dilated by a factor of 2 from the y axis then translated so that (x, y) -> (x – 3, y + 2). The equation of the image is:

Okay, here's the quick version of what we know, because I'm sleepy:

• Dilation by 2 from the y-axis
• We're shifted left by three units
• and we're shifted up by two units

As always, let's do the easy parts (translations, imo) first to see if we can skip some extra work and save time:

Going left three units means we have a positive h value because they're backwards. Let's slap that + 3 into our equation:

Bam, we've narrowed it down to A or B in two seconds. Let's add our k, +2:

Cool, so we can see that our answer is clearly A, as B pushes the graph down two units and that's not what we want. Don't need to work out the dilation, but you could if you wanted to; maybe while you're going back over your answers or something. It's up to you .

Feel free to yell at me if any of that is wrong, or I've made some silly typo that makes it impossible to comprehend. Good luck with your exams <3

Edit: Fixed Question 18; silly arithmetic error 
Edit: Fixed 17 too

[geminii]

And once again, you've saved my life!! Thank you so much, I've printed out your working out, looked through them, and I'll stick them in my bound reference

[MightyBeh]

I'm honoured! 

[geminii]

Oh the answer to question 18 was E

I think its because you did the translation last But no worries, I'll fix it up in my book

[MightyBeh]

Haha, I wasn't actually sure what E was meant to say so I didn't even consider it (don't do that on a SAC, obviously ). Is it mean to be ?

E y=(3x-)2 -4

[Callum@1373]

MightyBeh was wrong with Q17, the answer to that one is actually C.

It's actually asking the translation of y = (x-4)^2 + 5 to y = x^2

Not the translation of y = x^2 to y = (x-4)^2 + 5

So therefore it is 4 units to the right and 5 units above y = x^2 so it needs to go -4 units in the x axis and -5 units in the y axis.

Answer C is correct 

[geminii]

Hmmm I should let my maths teacher know, he was the one who told me the answer was B!!

Also, does anyone know how I would go about solving this?:

Question 11
Find a sequence of transformations which maps y=x2 to y= -5x2 +3.

Sorry it might look confusing, I don't know how to write the proper 'squared' signs!!

[MightyBeh]

Sorry it might look confusing, I don't know how to write the proper 'squared' signs!!

You can use this thing called LaTeX, it's the Pi sign when you're posting. It's a little bit complicated if you're doing anything with special signs but in general it's pretty simple. For example, to write , you'd do this:

Code: [Select]

[tex]x^2[/tex]

Question 11

Find a sequence of transformations which maps to .

This one's kind of like question 17; you can just apply it to the general form and fill in the blanks.





Here, it looks a bit different but it's definitely still applicable to the turning point formula:




Makes sense right? That means we can work in terms of a, h and k still. They're -5, 0 and 3 respectively. It says 'a sequence of transformations', which implies any order but it's good practice to describe using DR. T anyway (Dilation, reflection, translations).
There has been:

• a dilation by a factor of 5 from the x-axis;
• a reflection in the x-axis;
• and a translation three units in the positive direction of the y-axis.

That should be it, but I also threw it onto a graph if that makes it a little easier to catch.

Also throwing in a quick graphical explanation for why Q17 is C, not B:

Spoiler

We want to move the red one to the blue one, not the blue one to the red one. Or at least, that's what the question appears to say 

[geminii]

Thanks so much!! The graph really helps too