What is the period of the function f(x) = 4\cos\left(\frac{1}{3}x\right)? — ACT

What is the period of the function $f(x) = 4\cos\left(\frac{1}{3}x\right)$ ?

The period of the function is

6\pi

The standard period of the cosine function

y = \cos(x)

2\pi

. For a transformed trigonometric function of the form

y = A\cos(Bx)

, the period is given by the formula

\frac{2\pi}{|B|}

. In the function

f(x) = 4\cos\left(\frac{1}{3}x\right)

, the coefficient of

x

B = \frac{1}{3}

. Dividing the standard period

2\pi

\frac{1}{3}

yields a period of

2\pi \times 3 = 6\pi

Identify the standard period of the parent cosine function and the coefficient

B

of the variable

x

in the given function.

The parent function is

y = \cos(x)

, which has a standard period of

2\pi

. In the function

f(x) = 4\cos\left(\frac{1}{3}x\right)

, the coefficient of

x

B = \frac{1}{3}

This sets up the parameters needed for the period formula

\text{Period} = \frac{2\pi}{|B|}

Substitute the value of

B

into the period formula and simplify.

The period is

\frac{2\pi}{\frac{1}{3}} = 2\pi \times 3 = 6\pi

Dividing by a fraction is equivalent to multiplying by its reciprocal, which gives the final horizontal distance for one complete cycle.

Graphs of Trigonometric Functions

Question