Question

Write the equation of a sinusoidal function that rises from a minimum point at (-2,-1) to a maximum point at (3,9).

Answer 1

-- The difference between the maximum and minimum values (9 minus -1) = 10
is double the amplitude of the sine wave, so its amplitude is 5 .

-- The distance in 'time' from the minimum to the maximum (3 minus -2) = 5
is 1/2 of the cycle, so the "wavelength" is 10 .

-- The sine 'begins' halfway between the minimum and maximum = at (1/2, 4) ,
and it's 'phase' is proportional to ' x/10 '.
 
The generic sinusoid is Y = A sin(2π x) .

We know that A = 5 . And this particular wave also has a constant of 4 added to it.
Now I just have to pick my way through the argument of the sine.

Y = 4 + 5 sin [ 2π (x - 0.5)/10 ] .