Transformations

[Stick]

Now that we start using matrices for transformations, I'm really getting confused with going between function notation and matrix notation. I can apply the matrix method really well but I struggle when you need to transform one graph into another one when you're only given function notation. I'm not sure if I'm making any sense here. Are there any sure-fire methods or tips you have for these styles of questions?

Here's an example question so you get what I mean: g(x)=x^2-3x to 2g(x-1)-2

[brightsky]

yeah just remember that everything is the opposite of what it seems.
y = f(x-a) equates to a translation of a units in the POSITIVE direction of the x-axis (not negative, as it might seem to some people)
y - a = f(x) equates to a translation of a units in the POSITIVE direction the y-axis
y/a = f(x) equates to a dilation by a factor of a from the x-axis (not a dilation by a factor of 1/a from the y-axis as it might seem to some)
y = f(x/a) equates to a dilation by a factor of a from the y-axis
y = f(-x) equates to a reflection in the y-axis (NOT the x-axis it might seem)
-y = f(x) equates to a reflection in the x-axis (NOT the y-axis)

it will be beneficial for you to try and discover on your own using the maths you currently know why this is case. and take the time and effort to mull over how the matrix method and the 'inspection' method relate to each other. transformations are really important.

[Stick]

Thanks, brightsky! I knew that things are opposite (especially with the matrix method) but sometimes I still get a little muddled up. Perhaps if someone could show me how they would work out the example problem above and I might fully grasp it then. Bear in mind that I prefer using the matrix method to other methods.

[brightsky]

the matrix method is quite tedious and inefficient. y=g(x) --> y=2g(x-1) -2 would be:
1. dilation by a factor of 2 from the x-axis
2. translation of 1 unit in the positive direction of the x-axis
3. translation of 2 units in the negative direction of the y-axis
you should really be aiming to do this by inspection in a few secs.

[TrueTears]

Here's an example question so you get what I mean: g(x)=x^2-3x to 2g(x-1)-2

let y = g(x) and y'=2g(x'-1)-2 -> (y'+2)/2 = g(x'-1)

y = (y'+2)/2 and x = x'-1

rest is pretty obvious

[Stick]

Thanks, guys! Making a lot more sense now. I still prefer TT's method though (sorry, brightsky ).

How about this one? 1/x^2 to -3/(x-2)^2 +1

[brightsky]

okay, i'll explain the 'inspection' method in more detail (haha i feel like i'm spruiking wares). we shall apply the transformations one step at a time:
1. y=1/x^2 --> y=-1/x^2 (which is equivalent to -y = 1/x^2). so effectively, you are replacing y with (-y), which means a reflection in the x-axis.
2. y=-1/x^2 --> y = -3/x^2 (which is equivalent to -(y/3) = 1/x^2). now you are replacing y with (y/3), which means a dilation by a factor of 3 from the x-axis.
3. y = -3/x^2 --> y = -3/(x-2)^2. you are replacing x with (x-2), which means a translation of 2 units in the POSTIVE direction of the x-axis.
4. y = -3/(x-2)^2 --> y = -3/(x-2)^2 + 1 (which is equivalent to y -1 = -3/(x-2)^2). you are replacing y with y-1, which means a translation of 1 unit in the POSITIVE direction of the y-axis.

hope i'm making myself clear. :p

[pi]

I prefer brighsky's method tbh As long as you get it right I guess it doesn't matter.

[Jenny_2108]

Thanks, guys! Making a lot more sense now. I still prefer TT's method though (sorry, brightsky ).

How about this one? 1/x^2 to -3/(x-2)^2 +1

if you prefer TT's method
Let y= 1/x^2
and y'= -3/(x'-2)^2 +1
      y'-1 = -3/(x'-2)^2
     (y'-1)/(-3)= 1/(x'-2)^2
=> y=(y'-1)/(-3)
      (-3)y+1= y'
Dilation by a factor of 3 from x-axis, reflection in x-axis, translation 1 unit in positive direction y-axis

and x=x'-2 => x+2=x'
Translation 2 units in positive direction of x axis
 

[Stick]

Got it now. Thanks everyone.