Question

What is the phase shift of the function f left parenthesis t right parenthesis equals 4 cotangent left parenthesis 3 t minus Start Fraction 3 pi over 4 End Fraction right parenthesis plus 2?
(1 point)
Responses

left Start Fraction pi over 4 End Fraction
left Image with alt text: Start Fraction pi over 4 End Fraction

left Start Fraction 3 pi over 4 End Fraction
left Image with alt text: Start Fraction 3 pi over 4 End Fraction

right Start Fraction pi over 4 End Fraction
right Image with alt text: Start Fraction pi over 4 End Fraction

right Start Fraction 3 pi over 4 End Fraction

Answer 1

To find the phase shift for the function \( f(t) = 4 \cot(3t - \frac{3\pi}{4}) + 2 \), we can start with identifying the phase shift formula for cotangent functions.

The general form for the cotangent function is:

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

where:
- \( a \) is the vertical stretch,
- \( b \) affects the period,
- \( c \) is the phase shift,
- \( d \) is the vertical shift.

From the given function \( f(t) = 4 \cot(3t - \frac{3\pi}{4}) + 2 \), we can rewrite it in the form:

\[
f(t) = 4 \cot(3(t - \frac{\frac{3\pi}{4}}{3})) + 2
\]

Now, the term \( 3(t - \frac{\frac{3\pi}{4}}{3}) \) shows that:

\[
c = \frac{3\pi}{4} \div 3 = \frac{3\pi}{12} = \frac{\pi}{4}
\]

Since \( b = 3 > 0 \), the phase shift will be to the right:

\[
\text{Phase Shift} = \frac{\pi}{4}
\]

Thus, the phase shift of the function \( f(t) = 4 \cot(3t - \frac{3\pi}{4}) + 2 \) is:

\[
\boxed{\frac{\pi}{4}}
\]