Question

What is the phase shift of the function f(t) = tan (2t - 2pi/3) + 5 ?
(1 point)
right 5
right pi / 3
right 2pi/3
right 3pi/2
right 2pi/5

Answer 1

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

The standard form for a tangent function with a phase shift is:

\[
f(t) = a \tan(b(t - c)) + d
\]

where:
- \( a \) is the amplitude (not applicable for tan),
- \( b \) affects the period,
- \( c \) is the phase shift,
- \( d \) shifts the function vertically.

From your function, we have:

\[
\tan(2t - \frac{2\pi}{3}) = \tan(2(t - \frac{2\pi}{3} \cdot \frac{1}{2}))
\]

To express this in the standard form, we factor out the coefficient of \( t \):

\[
= \tan(2(t - \frac{\pi}{3}))
\]

In this expression, \( b = 2 \), and the phase shift \( c \) can be found from \( t - \frac{\pi}{3} \).

To find the phase shift, we can set:

\[
t - \frac{\pi}{3} = 0 \implies t = \frac{\pi}{3}
\]

This means the phase shift is to the right by \( \frac{\pi}{3} \).

So the answer is:

**right \( \frac{\pi}{3} \)**.