Calculus Question. Please help!

Question

 

Write the equation of the cosine function that models the values in the following table.

x 0 5 10 15 20 25 30

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y -5 -8 -11 -8 -5 -8 -11

Answer 1

The generic cosine function is f(x) = Acos(B(x - C)) + D where:

A = the amplitude (distance from the midline to the max or min)

B = 2π/period

C = horizontal shift

D = vertical shift

Plotting the points gives this picture:

and you can get everything you need from it.

The midline is at y = -8 meaning D = -8

Since the distance between the midline and either the max or the min is 3 then A = 3

Since the function begins repeating after 20 units, the period is 20 so B = 2π/20 or π/10

The function starts at the top when x = 0 just like a cosine function should so there is no horizontal shift, or C = 0

Answer 2

ω = 2π/20 = π/10

Then y = − 8 + 3 cos ( πt / 10 )