sin and cosine equations

Question

A cosine curve with a period of 4𝜋, an amplitude of 3, a right phase shift of 𝜋/2, and a vertical translation up 5 units.

Answer 1

The generic cosine function is:

where "A" is the amplitude, "B" defines the period where the period is 2π/B, "C" is the horizontal shift or phase shift, and "D" is the vertical shift or midline location.

We can just plug in A = 3, C = π/2 and D = 5 but we need to calculate "B". since the period is 2π/B then B = 2π/period so B = 2π/(4π) = 1/2

Notice that for the phase shift it is always opposite of the way it looks, so cos[1/2(x - π/2)] means it shifts RIGHT by π/2

Answer 2

y = 3Cos[(1/2)(x - pi/2)] + 5

In general, y = aCosb(x-c) + d

a = amplitude = 3

b = 2pi/period = 2pi/4pi = 1/ 2

c = phase shift = -pi/2 (to the right is considered to be negative)

d = vertical shift = 5

Note that the high will be 8 and the low will be 2.