For d - if I’d have done the translation first ( you never covered it in the video) , it would be (0,-2) right ? Because when you multiply three by that one three, you’ll stick a three in front of the f(x) and three times ‘-2’ to get your minus 6?
for (d) i did translation (0, -2) then stretch in y direction by SF 3, does that get the same answer or is it only the X that is affected by the stretch?
Hey Sir, Is there a generic rule for the order of transformations? What I seem to to understand is that if it doesn't affect the same axis then order doesn't matter, however if the transformations affect the y-axis then the order is usually stretch then translate, however when it only affects the x-axis then it's translation then stretch...is that right?
Selena G I know you're probably looking for an easy way out here, but when a transformation affects the same axis then although the order 'matters', this might just mean you can reverse the order but that the translation vector or stretch factor might be different. For example, let's say we want to describe a sequence of two transformations that transforms y=x^3 to y=(2x+1)^3. If we do a translation first by the vector [-1,0], then we first get y=(x+1)^3. Then we do a stretch parallel to the x-axis by factor 1/2, then we get y=(2x+1)^3. Instead, if we first do the stretch parallel to the x-axis by factor 1/2, then we get y=(2x)^3. We can then do a translation, but if we do it by the vector [-1,0] as before, then we get y=(2(x+1))^3, which is equal to y=(2x+2)^3, which isn't what we want. Instead, we can do a translation by the vector [-1/2,0], so that we get y=(2(x+1/2))^3, which is equal to y=(2x+1)^3, as required. So really we can do a translation or a stretch in either order, even in the same axis, but you just have to be careful with the vector and/or stretch factor that you use.
Both work vertically. Let's say you start with the coordinates (1,0). Translate by 1 vertically to get (1,1). Then reflect in the x-axis, you get (1,-1). If you reflect in the x-axis first, you get (1,0). Then translate by 1 vertically you get (1,1). Because we've ended up at different points, the two affect one another.
It doesn't matter if you do the inside ones first, or the outside ones first. For those on the outside, do the a first then the d. For those on the inside, do the c first then the b.
On the 2nd example, how come when doing a stretch of sf 1/3 in the y-direction you put 3y first? Is that just the way you've been taught it? Because multiplying by 1/3 straight away gets you the same answer. Thanks
It's using replacements, which is what you need to think about if a curve is written implicitly. For example, if you had the circle x^2 + y^2 = 1, you can apply a stretch by factor 1/3 in the y-direction by replacing y with 3y: x^2 + (3y)^2 = 1, giving you an ellipse. This replacement method I often use in general to explain stretches.
No. Let's say you were going to do a stretch by factor 1/3 in the y-direction to y = x^2, then you would get y = (1/3)x^2. This is equivalent to 3y = x^2 (hence replacing y with 3y).
It depends how you learn it. If you learn it so that when you stretch in the x-direction by factor k, you replace x with (1/k) x, and if you stretch in the y-direction by factor k, you replace y with (1/k) y, then you're able to perform transformations on curves that are written implicitly. For example, if you wanted to stretch the circle x^2 + y^2 = 25 by 1/3 in the x-direction and 1/5 in the y-direction, then replace the x with 3x and the y with 5y and you get: (3x)^2+(5y)^2=25, which is now the transformed shape (an ellipse).
way less confusing too because you're using the sf straight away. doing the reciprocal in your head isn't hard it's just less effort to not consider it
