The graph of a sinusoidal function has a maximum point at (0,10) and then intersects its midline at (pi/4,4). Write the formula of the function, where x is entered in radians. f(x)=

There is a family of sinusoidal functions of the form:

Where is associated to the angular frequency of a function within the family.

In this case, the sinusoidal function reaches the upper bound when and the middle value when . We should find all coefficients contained in the following sinusoidal model:

(1)

Where:

If we know that , , and , then we have the following system of equations:

(2)

(3)

By (2):

By (3):

,

,

There is a family of sinusoidal functions of the form:

(4)

Where is associated to the angular frequency of a function within the family.

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y = 6 cos (2x) + 4

Step-by-step explanation:

A cosine function is expressed as: y = A cos (Bx - C) + D   where

A = Max - Midline

   = 10 - 4

   = 6

The max to the midpoint is π/4

⇒ 1/4 P = π/4

⇒ P = π

B = 2π/P    

  = 2π/π

  = 2

C = 0 because there is no phase shift

D = 4 (given)

Input A = 6, B = 2, C = 0, and D = 4 into the cosine function:

y = 2 cos (2x - 0) + 4