Precalculus homework

Question

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

x 0 4 8 12 16 20 24

y 3 -1 -5 -1 3 -1 -5

'y=

Answer 1

Need to fit the points to

y = Asin(B(x+C))+D

The period is 16: B = 2π/p = π/8

D = (max+min)/2 = (3-5)/2 = -1

sin(π/2) = 1 so when x = 0, BC = π/2 --> C = (π/2)(8/π) = 4

A(1)+D = 3 = A-1 --> A=4

y = 4sin(π(x+4)/8) -1

Checking:

y(0) = 4-1 = 3

y(4) = 4sin(π)-1 = -1

y(8) = 4sin(3π/2)-1 = -4-1 = -5

y(12) = 4sin(2π)-1 = -1

y(16) = 4sin(5π/2)-1 = 4-2 = 3

y(20) = 4sin(4π)-1 = -1

y(24) = 4sin(7π/2)-1 = -4-1 = -5

Answer 2

The cycle is length 16 (when x is 0 and 16, y = 3; when x is 8 and 24, y = -5; the min is -5 and the max is 3, so the amplitude is (5 - -3)/2 = 4;the displacement is (5 + -3)/2 = 1.

Thus, the angular frequency is 2π / 16 = π / 8.

The maximum is at 0.

Thus, the function is

y = 4 cos(xπ / 8) + 1