For of the periodic function given in the graph below state

For of the periodic function given in the graph below, state the values of the period, the phase shift, the vertical shift, and the amplitude, then write the equation in the form: f(x) = c + a cos [b(x - d)] period: _____ phase shift: _____ vertical shift (midline): _____ amplitude: _____ f(x) = _____

Solution

Base function - cos (x)

f(x) = c + a cos [b (x-d)]

18. period : The x value after which the values of f(x) starts repeating

at x = 0, f(x) = 5

at x = 0.5, f(x) = 5

so period is 0.5

19. Phase shift (d): difference in the input of f(x) from base function (cos x)

at x = 0, cos x = 0

here too at x =0, f(x) = 0

so no phase difference. phase difference (d) = 0

20. Vertical Shift (c) : shift upwards on the f(x) values from base function

midline of cos x : y = 0

midline of f(x) : y = 0.5 (5+ (-3)) = 1

so vertical shift is 1.

21. Amplitude (a)

at x = 0, f(x) = 5, but 1 is due to shift so effective amplitude is 4

so amplitude is 4.

(it can also be checked at minimum value of f(x) and cos (x). Min (f(x)) = -4 + 1 = -3 (1 is due to vertical shift)

22. f(x)

f(x) = 1 + 4 cos [ 4 (x)]

at x = 0, b(x-d) = 0

at x = 0.125, b(x-d) = /2

at x = 0.25, b(x-d) =

at x = 0.375, b(x-d) = 3/4

at x = 0.5, b(x-d) = 2