What is the general equation of a cosine function with an amplitude of 3, a period of 4 pi, and a horizontal shift of -pi?

The parent function is y = cos x, which has amplitude of one and period of 2π (Refer to the first diagram attached below)

To increase the amplitude to three, we will have to stretch the graph vertically.
The function f(x) then become f(x) = 3 cos (x) 
(Refer to the second diagram)

To increase the period from 2π to 4π, we stretch the graph horizontally.
The function f(x) then become f(x) = 3 cos (0.5x)
(Refer to the third diagram)

Lastly, to shift the graph by π to the left, we translate the graph and the function is f(x) = 3 cos (0.5x + π)

Cosine function have general form for which amplitude, period and phase shift can be specified. The needed equation is y = 3cos(x/2 + π/2)

What is the general equation of a cosine function?

Suppose that the cosine function has:

Amplitude = 'a'Period = Phase shift(horizontal)(left) = c/b

Then, its equation is given as:

y = a.cos(bx +c)

For the given context, we have:

Amplitude = a = 3

Period= 4pi

and Horizontal shift = π(since -pi is left and we need only magnitude of shift to left)

Since period = , thus, getting value of b:

= 4π, b = 1/2

And now, as horizontal shift = c/b = π, thus, c = bπ = π/2

Thus, the cosine function needed is given as:

y = a.cos(bx+c) = 3cos(x/2 + π/2)

Its graph is plotted below, with the standard graph of y = cos(x) for comparison.

Thus,

The needed equation is y = 3cos(x/2 + π/2)

Learn more about general cosine function here:

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