VCE Methods Question Thread!

[Shadowxo]

Hi everyone! I was wondering if anyone could help me solve this question:

Consider the simultaneous linear equations:

mx - 6y = 6
4x - my = m

Find the values of m for which the equations have infinitely many solutions.

My teacher said we should put the equation into matrix form, which i have done as follows:

[m   -6]

•   =  [6]
[4   -m]

[y]  =  [m]

My teacher said we then need to use the determinant (ad-bc), which in this case would be:

m^2 - (4)(-6)
= m^2 + 24.

But I am not sure whether the determinant should be less than 0, equal 0 or be greater than 0? I know that if there are no solutions, the determinant should = 0. But what about for infinitely many solutions?

Thank you so much! :D

I've never done it via matrices. One thing you could do is multiply them by either 6 or m so they equal the same value (6m) and equate from there?
The way I was taught was to first put them in the form y=mx+c and if there are infinitely many solutions, the gradient and y intercept (m and c respectively) are equal
y= (m/6)x -1 and
y= (4/m)x -1
The y intercepts are already the same, so m/6 = 4/m => m²=24 => m=2√6

[Shadowxo]

http://m.imgur.com/a/Mxg0k

Q5) I'm a bit confused, if when we dilate by a factor in the y-axis, do we apply it to everything inside the function. In this case (x+12), this seems true since you get the correct answer if you do so.

However, if we reflect in the the y-axis would this also mean we multiply (x+12) by -1? My teacher did an example with a similar question, but when he reflected in y, he did not change the sign of the constant inside of the function, he only did it to x.

Q4f) I did this both algebraically and by observation, however if I do it by observation it contradicts what I wrote in the image. The problem arises (kind of relates to the problem in Q5) after I translate in x and dilate. If I reflect in y (as in multiply entire inside function by -1), I get x-3. Wouldn't this suggest that I would have to translate 3 units negatively in the x-axis instead of positive (as seen in the image)?

Cheers

For dilations / translations / reflections you only do it to the x or y.
So when dilating by a factor of 2 from the x axis (doubling the y value), you multiply the whole function by 2. When dilating by a factor of 3 from the y axis (tripling the x value) you change x to x/3. Any constants etc you ignore. This is also why order is important, dilating by a factor of 4 then translating 3 units is different to doing it the opposite order.
So for 5, you only affected the x value and ignored the 4.

So if you reflect in the y axis (affecting the x value) for x+12, you only change the x, so it would change to -x +12. When reflecting in the x axis you multiply the whole function by -1.

4.
You only affect the x or y, ignore the constants when doing transformations.

Hope this helps

[Syndicate]

Hi everyone! I was wondering if anyone could help me solve this question:

Consider the simultaneous linear equations:

mx - 6y = 6
4x - my = m

Find the values of m for which the equations have infinitely many solutions.

My teacher said we should put the equation into matrix form, which i have done as follows:

[m   -6]

•   =  [6]
[4   -m]

[y]  =  [m]

My teacher said we then need to use the determinant (ad-bc), which in this case would be:

m^2 - (4)(-6)
= m^2 + 24.

But I am not sure whether the determinant should be less than 0, equal 0 or be greater than 0? I know that if there are no solutions, the determinant should = 0. But what about for infinitely many solutions?

Thank you so much! :D

When determinant equals 0, the gradient of the lines can either yield a system of linear equations that are consistent or inconsistent. You just need to substitute the m-value to determine which one yields a system with no solution, or the one with infinitely many solutions.

[geminii]

When determinant equals 0, the gradient of the lines can either yield a system of linear equations that are consistent or inconsistent. You just need to substitute the m-value to determine which one yields a system with no solution, or the one with infinitely many solutions.

Oh I get it, thank you so much!!!!

[deStudent]

For dilations / translations / reflections you only do it to the x or y.
So when dilating by a factor of 2 from the x axis (doubling the y value), you multiply the whole function by 2. When dilating by a factor of 3 from the y axis (tripling the x value) you change x to x/3. Any constants etc you ignore. This is also why order is important, dilating by a factor of 4 then translating 3 units is different to doing it the opposite order.
So for 5, you only affected the x value and ignored the 4.

So if you reflect in the y axis (affecting the x value) for x+12, you only change the x, so it would change to -x +12. When reflecting in the x axis you multiply the whole function by -1.

4.
You only affect the x or y, ignore the constants when doing transformations.

Hope this helps

Thanks!

Lastly for this http://m.imgur.com/a/fdTdu

Q3c) can someone check if my working is correct? It seems correct algebraically but just want to double check. Although the +15 in y-axis looks dodgy by looking at the equations..

Q4c) I'm checking my answer by observation, but shouldn't there be a 1 unit negative translation in the x-axis, instead of positive? It contradicts the algebraic method..

[Shadowxo]

Thanks!

Lastly for this http://m.imgur.com/a/fdTdu

Q3c) can someone check if my working is correct? It seems correct algebraically but just want to double check. Although the +15 in y-axis looks dodgy by looking at the equations..

Q4c) I'm checking my answer by observation, but shouldn't there be a 1 unit negative translation in the x-axis, instead of positive? It contradicts the algebraic method..

Your working for both questions is right
For 4c), you can either reflect in y axis and translative 1 unit in positive direction of the x axis OR translate 1 unit in negative direction of x axis then reflect in y axis, just depends on the order you do it!

[Joseph41]

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[deStudent]

http://m.imgur.com/a/fmqJv

Q10) I did this fine algebraically, I just had to not expand g(f(x)). But when doing it by observation, when you look at g(f(x)), if you dilate by a factor 1/3 from the y-axis, it looks like you just need to translate 2 units in the positive direction of x after, instead of 2/3 units. Why is that? Since you're dilating first, shouldn't the translation of 2 units, remain as 2 units since it goes after?

Q2f) Their graph of f(x) is wrong right? It's supposed to be a square root graph, but it looks like they drew a cubic? Also, is anyone else having problems finding the other points of intersection between these graphs apart from the one on y=x? It doesn't work for unless I plot it, but then it'll be in decimal form..

Thanks

[vasuk]

Hello can someone assist me in this pls
When you get questions like these, for example in part c, do you find the COMPOSITE f∘h first or do you find the h(2) first?

[Shadowxo]

http://m.imgur.com/a/fmqJv

Q10) I did this fine algebraically, I just had to not expand g(f(x)). But when doing it by observation, when you look at g(f(x)), if you dilate by a factor 1/3 from the y-axis, it looks like you just need to translate 2 units in the positive direction of x after, instead of 2/3 units. Why is that? Since you're dilating first, shouldn't the translation of 2 units, remain as 2 units since it goes after?

Q2f) Their graph of f(x) is wrong right? It's supposed to be a square root graph, but it looks like they drew a cubic? Also, is anyone else having problems finding the other points of intersection between these graphs apart from the one on y=x? It doesn't work for unless I plot it, but then it'll be in decimal form..

Thanks

10) Order matters. x -> 3x+2. Either you dilate by 1/3 from y axis then translate by 2/3, or translate by 2 and dilate by 1/3. For this it's not a regular function, f(x), you're changing the x value. If you translate first then dilate after, the value of the translation will be 1/3 of the value, as it's dilated.
2f) their graph is terrible, it looks like a cubic but it's not. The other solutions should be findable using graphics calculator, using the solve feature.

Hello can someone assist me in this pls
When you get questions like these, for example in part c, do you find the COMPOSITE f∘h first or do you find the h(2) first?

It's f(h(2)), so find h(2) then sub it in to find f(h(2))

[deStudent]

Thanks Shadowxo.

For some reason I still can't get my head around +2/3 on my question. If we dilate first meaning we're dealing with x first right? This means x->3x, since its first it does not affect the translation at all right? So my brain keeps thinking that I just need to +2.

Pretty confused how 2 goes to 2/3 when the dilation is before the translation..

[Shadowxo]

Thanks Shadowxo.

For some reason I still can't get my head around +2/3 on my question. If we dilate first meaning we're dealing with x first right? This means x->3x, since its first it does not affect the translation at all right? So my brain keeps thinking that I just need to +2.

Pretty confused how 2 goes to 2/3 when the dilation is before the translation..

It's good to imagine it.
If you have a graph, if you translate it left a bit then shrink it (dilate by factor of 1/n eg 1/3 in direction of x axis), that translation matters less. If you dilate first, the translation is bigger. eg - If you start off at a point where x=3, translating it by 1/2 would change it to be 5/6ths of it's original value, then the dilation would make it 1/3 of that - ends up at 5/6. If you dilate first, that makes it 1 (1/3 of the original value), then translating it by 0.5 would decrease it by half, making it 1/2 (almost half of the other way around).
Best way I can explain this is using examples

[Gogo14]

For this question, I got 3/2 while the answers got 12. Confused

[Mattjbr2]

Hi, would someone be able to tell me the difference between the words "across, about, in, on, through" when it comes to graph reflections? It seems as though no one can come to an agreement as to what term they use. In one book i see "reflect about the y axis" in another i see "reflect across the y axis". I tried researching this question but it also seems as though I'm the only one confused in this regards. Do they all mean the same thing or?

[Mattjbr2]

For this question, I got 3/2 while the answers got 12. Confused

Here u go brah. Sorry for the crappy formatting, did it on a crowdy train for fun