How do I find a sinusoid equation with a maximum value of 5 and a minimum value of 1 that intercepts through points (3,1) and (5,3)?

Answer 1

If max=5 and min=1, average these to find the midline: (5+1)/2 = 6/2 = 3. So the general equation is:

y=Asin(B(x+C))+D. This is "D". The A is the amplitude. The amplitude is the positive value for the distance from the midline to the max/min. Here it is 2 (from 3 to 5 = 2, from 1 to 3 = 2)

Now we have y=2 sin(B(x+C) + 3

You have a point at a minimum (3,1) and a point that passes through the midline (5,3). This is one quarter of the cycle, and it lasts 2 units, so the full period would be 8. B = 2pi/T, so 2pi/8 = pi/4

Now we have y = 2 sin[ (pi/4) (x +C) ] + 3

If you graph this accurately, you will see that the sin cycle starts back 3 units, so C = 3, and the final

equation is

y=2 sin[ (pi/4)(x+3) ] +3

Or as a cosine, it starts back just 1 unit,

y= 2 cos[ (pi/4)(x+1) ] +3

Both of these are shown in the graph below.

https://www.desmos.com/calculator/rrodjqblb5