What is the cosine function for a period of 6 and a maximum of 3 at x=-2 and a lowest point of -1?

Question

Hello, I would kind of need a detailed solution here since I'm advance studying for our trigonometry class thank you!

Answer 1

What is the cosine function for a period of 6, a maximum of 3 (at x = – 2) and a minimum of -1 (lowest point)?

General form of cosine function:

y = acos(k(x – b)) + h

where:

amplitude = |a|

period = 2pi/k

phase shift = b

vertical shift = k

if a > 0 then function value is maximum at x = b

if a < 0 then function value is minimum at x = b

Problem iformation:

Period = 6

Maximum = 3 @ x=-2; point (-2,3)

Minimum = -1

Amplitude

Function range = Max. – Min.  = 3 – (-1) = 4

Amplitude = ½ Range

|a| = ½(4) = 2

a = 2 (assuming a > 0; maximum value is at x = b)

Maximum value occurs at x = -2

b = -2

Vertical shift

h = Vertical shift (midline of graph) = -1 + 2 = 1

Period

Period = 2pi/k

k = 2pi/period

k = 2pi/6

k = pi/3

y = 2cos(pi/3(x –(-2)) + 1

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