Question

The graph of f(x)=sin(x) is transformed into a new function, g(x), by stretching it vertically by a factor of 4 and shifting it 3 units to the right.

What is the equation of the new function g(x)?

Answer 1

if you stretch it out by a factor of 4, it means that the x values change only 1/4 as fast.
sin(1/4 x) means you now have to have 4 times the value of x to get to the same place.
Similarly, you have to subtract 3 from the new x to get the original y.
So,
g(x) = sin(1/4 (x-3))

you can see the results here:

https://www.wolframalpha.com/input/?i=plot+y%3Dsinx,+y%3Dsin(1%2F4+(x-3))

Answer 2

But y = sinx stretched vertically by a factor of 4 and shifted 3 units to the right would be
y = 4sin(x-3)

see:

Answer 3

see:
https://www.wolframalpha.com/input/?i=plot+y%3Dsinx,+y%3D+4sin(x-3)