SOLUTION: A sinusoidal function has an amplitude of 8 units, ad period of 180, and a minimum at (0, -3). Determine the equation of the function.

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Question 1202328: A sinusoidal function has an amplitude of 8 units, ad period of 180, and a minimum at (0, -3). Determine the equation of the function.
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52861)   (Show Source): You can put this solution on YOUR website!
.
A sinusoidal function has an amplitude of 8 units, ad period of 180,
and a minimum at (0, -3). Determine the equation of the function.
~~~~~~~~~~~~~~~~ 

Since this sinusoidal function has a minimum at x= 0, it tells us
that the pattern is  m-a*cos(bx)  without a phase shift,

where "m" is the midline,  "a" is the amplitude and "b" is the coefficient 
managing the period.


About the amplitude, we are given that a= 8;  hence, m = 8-3 = 5.


Since the period is 180 units, we have

     = 180,

so  b =  = .


Thus and finally,  the function is  f(x) = .    ANSWER

Solved.



Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: y = -8cos(2x)+5
This is when we are in degree mode.
Other answers are possible.

================================================================================

Explanation:

The template for cosine is
y = A*cos(B(x-C))+D

where,
|A| = amplitude
B = used to find the period
C = phase shift
D = midline

|A| = 8 leads to either A = 8 or A = -8
Let's go with A = -8 because A = 8 will make x = 0 lead to a max, when we want x = 0 to lead to a min instead.

"period of 180" seems to imply your teacher wants things in degree mode (rather than radians).
T = 180 = period
B = 360/T
B = 360/180
B = 2

To make things simple, I'll have the phase shift be set to C = 0.

One minimum point is located at (0,-3)
The smallest y output possible is y = -3
Go up 8 units, the amplitude amount, to arrive at y = -3+8 = 5 which is the midline
Therefore, D = 5


Summary:
A = -8
B = 2
C = 0
D = 5

y = A*cos(B(x-C))+D
y = -8*cos(2(x-0))+5
y = -8cos(2x)+5

Graph
https://www.desmos.com/calculator/gigkk7becm
GeoGebra is another graphing option I recommend.
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