Which of the following values represents the period, in radians, of the trigonometric func… — ACT

Which of the following values represents the period, in radians, of the trigonometric function $f(x) = 3 \tan\left(\frac{2}{3}x - \frac{\pi}{6}\right) + 5$ ?

$\frac{3\pi}{2}$ Answer
B
$3\pi$
C
$\frac{2\pi}{3}$
D
$\frac{\pi}{6}$
E
$\frac{3\pi}{4}$

Answer

The period of the given tangent function is

\frac{3\pi}{2}

radians.

The parent function

y = \tan(x)

repeats every

\pi

radians. For a function in the form

f(x) = A \tan(Bx - C) + D

, the horizontal scale factor

B

alters the period according to

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

. With

B = \frac{2}{3}

, dividing

\pi

\frac{2}{3}

gives

\frac{3\pi}{2}

Step-by-Step Solution

Identify the standard form of the transformed tangent function and its parameters.

For

f(x) = A \tan(Bx - C) + D

, the coefficient of

x

B = \frac{2}{3}

The horizontal stretch/compression factor

B

determines the period of the function.

Apply the period formula for the tangent function.

\text{Period} = \frac{\pi}{|B|} = \frac{\pi}{\frac{2}{3}} = \frac{3\pi}{2}

Unlike sine and cosine functions which have a fundamental period of

2\pi

, the parent tangent function

y = \tan(x)

has a period of

\pi

radians.

Key Concept

Period of Transformed Tangent Functions

Estimated Time:1m 0s

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