can you explain to me the transformations in sinusoidal functions especially the vertical compression/stretch and horizontal compression/stretch?

Answer 1

for example
y = a sin k x
where a and k are constants that we can fool with?

If you increase a, you increase the height of the function. If a is 1, then the function all happens between y = -1 and y = +1
If a is 2, then y swings back and forth between -2 and +2
etc

In the horizontal or x direction, the function repeats every time kx changes by 2pi
so if kx = 0, y = 0 and increasing
then when kx = 2pi, y = 0 and increasing again
that is when x = 2 pi/k
so the "wavelength", the x distance over which the function repeats, is 2 pi/k
so it