Question

What is the phase shift of the function f left parenthesis t right parenthesis equals tangent left parenthesis 2 t minus Start Fraction 2 pi over 3 End Fraction right parenthesis plus 5?
(1 point)
Responses

right 5
right 5

right Start Fraction pi over 3 End Fraction
right Image with alt text: Start Fraction pi over 3 End Fraction

right Start Fraction 2 pi over 3 End Fraction
right Image with alt text: Start Fraction 2 pi over 3 End Fraction

right Start Fraction 3 pi over 2 End Fraction
right Image with alt text: Start Fraction 3 pi over 2 End Fraction

right Start Fraction 2 pi over 5 End Fraction

Answer 1

To determine the phase shift of the function \( f(t) = \tan(2t - \frac{2\pi}{3}) + 5 \), we need to rewrite the tangent function in the standard form \( f(t) = \tan(k(t - d)) + C \), where \( d \) is the phase shift.

1. Identify \( k \) and the expression inside the tangent function:

The function can be expressed as:
\[
f(t) = \tan(2(t - \frac{\pi}{3})) + 5
\]

Here, \( k = 2 \) and the phase shift \( d \) can be found by factoring \( 2 \) out:
\[
2t - \frac{2\pi}{3} = 2(t - \frac{\pi}{3})
\]
Thus, \( d = \frac{\pi}{3} \).

2. Since the tangent function typically has a phase shift of \( d \) to the right if \( d \) is positive, we have:

The phase shift is \( \frac{\pi}{3} \) to the right.

Therefore, the correct answer is:
\(\frac{\pi}{3}\)