Question

If a sinusoidal function has a local maximum at (3,8) and the next local minimum at (7,-2),

1) What is the equation of a cosine function that has a graph characterized in the statement above
2) What is the equation of a sine function that has a graph characterized in the statement above

Answer 1

y = a cos (k x - b) + c
a and c are easy because c is average of 8 and -2 or 3 and a is thus 5
y = 5 cos (k x - b) + 3

cos is max of 1 at k x - b = 0
k*3-b = 0
so
b = 3 k
so y = 5 cos (k x - 3 k) + 3

cos is min of -1 at k x - b = pi
kx - 3k = pi but x = 7 here so
7 k - 3 k = pi
k = pi /4
so now
y = 5 cos pi/4( x - 3 ) + 3
or y = 5 cos (pi x/4 - 3 pi/4)
now you do it for sine wave :)