find the period and equation of a function

Question

a. the average monthly temperature in New York varies from a minimum of 43 degree in January to 89 degrees in July. Let T(m) be the average high temperature in a given month, with m being months past January of 2019

1. what is the period of T(m)?
2. what are the first four solutions of the equation T(m)=66?
3. find an equation for T(m)

Answer 1

For T(m) = asin [b(x + c)] + d

a = |0.5(89 - 43)| = 23

period = 12 months = 2pi/b therefore b = 2pi/12 = pi/6

d = midline is at T(m) = 0.5(89 + 43) = 66

The sine function needs to shift to the right one-fourth of the period for lowest point (0, 43) to be the y-int.

c = 0.25(b) = 3 and shift to right requires - 3

T(m) = 23sin[pi/6(m - 3)] + 66