Find a possible formula for the trigonometric function whose values are in the following table.

Question

x: 0,2,4,6,8,10,12

y:9,3,-3,3,9,3,-3

y=?

Answer 1

1) the max value is when x = 0 which matches a cosine function.

y = Acos(B(x - C)) + D where A is the amplitude, B is 2π/period, C is the horizontal shift, and D is the vertical shift.

2) There are evenly spaced x-values and the y-values oscillate between -3 and 9. The average between -3 and 9 is (-3 + 9)/2 which gives the midline (aka, vertical shift). So D = (-3 + 9)/2

3) The amplitude is the y-distance between the midline and the max (or min). So A = 9 - (-3 + 9)/2

4) The period is the distance in the x-direction until the sinusoid repeats. The distance between 9 and the next 9 is (8 - 0) so B = 2π/(8 - 0)

5) There is no shift in the horizontal direction because, as mentioned, the function starts at its max and goes down from there which is a standard (non-shifted) cosine. So C = 0.

Answer 2

Need to fit the points to Acos(B(x+C))+D.

I picked cos() since range of cos() is [-1,1].

Acos(0) + D = A+D=9, the maximum value.

D = (max+min)/2 = (9-3)/2 = 3. Which makes A = 6.

The values repeat at x=8, so the period is 8. B = 2π/period = π/4

Half a period is 4.

For C, π/4+C = 0 and (π/4)(4)+C = π to get the maximum and minimum values, so C = 0

y(x) = 6cos(πx/4)+3

Checking

y(0) = 6(1)+3 = 9

y(2) = 6(0)+3 = 3

y(4) = 6(-1)+3 = -3

y(6) = 6(cos(3π/2))+3 = 6(0)+3 = 3