How to change amplitude, phase and period of $x^2\sin(2x)$?

Question

given $y=x^2\sin(2x)$, with x between $-2\pi$ and $2\pi$, I need to make the amplitude $4$ times larger, shift the phase $1$ to the right and double the width. With trial and error, I could find that $(x+1)^2\sin(x+1)$ gives the equation I need, but how can I get there mathematically?

Answer 1

Period: When $x$ inside $\sin(x)$ gets to $2\pi$, that's one full phase. So if you instead use $\sin(x/2)$ now when you plug in $x = 2\pi$, you end up evaluating $\sin(\pi)$ which is only half a phase, so if you divide the argument of sin by 2 it will take twice as long to reach $2\pi$, so the period will be twice as large.

Phase: Consider $\sin(x+1)$, when you evaluate this at $x = 0$, you end up calculating $\sin(1)$, so the value that you would usually find at $x = 1$ for $\sin(x)$ is now found at $x = 0$, so it's a shift to the left.

Amplitude: Consider $5\sin(x)$, this will take whatever the result of $\sin(x)$ is and then multiply by $5$, so the height (amplitude) will be 5 times as large as the amplitude of $sin(x)$