Need Help with some questions (transformation)

[lakvinu]

 Hello,

im was doing some practise exams and i just need some help. So A(x)= e^x * (sin(x))^2 and V(x)=  e^-x * (sin(x))^2. It asking the transformation from A(x) to V(x). Im good at transformation but i dont know how to change the x without replacing the sin one as well. I thought that um sin(x) and sin(-x) would be the same so u dont have to worry about that but im not sure. can someone pls tell me how to do it. All i can think of is reflecting in the y axis but wouldnt that change the sin(x) as well. Pls Help thx.

Also another transformation question.
f(x)=sin(2x). g(x)=2 cos(x). How do i go from f(x) to g(x).

Thanks

[S_R_K]

Hello,

im was doing some practise exams and i just need some help. So A(x)= e^x * (sin(x))^2 and V(x)=  e^-x * (sin(x))^2. It asking the transformation from A(x) to V(x). Im good at transformation but i dont know how to change the x without replacing the sin one as well. I thought that um sin(x) and sin(-x) would be the same so u dont have to worry about that but im not sure. can someone pls tell me how to do it. All i can think of is reflecting in the y axis but wouldnt that change the sin(x) as well. Pls Help thx.

Your hunch is right, but your reasoning is not quite there. The thing to notice here is that:



Hence, V(x) is the image of A(x) under the transformation where (x, y) gets mapped to (–x, y) – as you say, a reflection in the y-axis.

In other words, it is not true that sin(x) = sin(–x). Rather, we have sin(x) = –sin(–x). But when you square both sides, each is positive. Hence sin^2(x) = sin^2(–x).

Also another transformation question.
f(x)=sin(2x). g(x)=2 cos(x). How do i go from f(x) to g(x).

I assume you know a sequence of transformations that maps sin(2x) to 2sin(x). To map from sin(x) to cos(x), notice that cos(x) is the image of sin(x) under a translation along the x-axis. Sketching a graph of each of sin(x) and cos(x) on the same set of axes should help you see exactly what this translation is, and hence how to transform sin(x) to cos(x). Then just find the dilations that give you the right amplitude / period.

Let us know if you need further hints.