Find Amplitude, Period, and Phase Shift f(t)=4cot(3t-(3pi)/4)+2

$f (t) = 4 \cot (3 t - \frac{3 π}{4}) + 2$

Step 1

Use the form $a \cot (b t - c) + d$ to find the variables used to find the amplitude, period, phase shift, and vertical shift.

$a = 4$

$b = 3$

$c = \frac{3 π}{4}$

$d = 2$

Step 2

Since the graph of the function $c o t$ does not have a maximum or minimum value, there can be no value for the amplitude.

Amplitude: None

Step 3

Find the period using the formula $\frac{π}{| b |}$ .

Find the period of $4 \cot (3 t - \frac{3 π}{4})$ .

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

Find the period of $2$ .

The period of the function can be calculated using $\frac{π}{| b |}$ .

$\frac{π}{| b |}$

Replace $b$ with $3$ in the formula for period.

$\frac{π}{| 3 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{π}{3}$

The period of addition/subtraction of trig functions is the maximum of the individual periods.

$\frac{π}{3}$

Step 4

Find the phase shift using the formula $\frac{c}{b}$ .

The phase shift of the function can be calculated from $\frac{c}{b}$ .

Phase Shift: $\frac{c}{b}$

Replace the values of $c$ and $b$ in the equation for phase shift.

Phase Shift: $\frac{\frac{3 π}{4}}{3}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{3 π}{4} \cdot \frac{1}{3}$

Cancel the common factor of $3$ .

Factor $3$ out of $3 π$ .

Phase Shift: $\frac{3 (π)}{4} \cdot \frac{1}{3}$

Cancel the common factor.

Phase Shift: $\frac{3 π}{4} \cdot \frac{1}{3}$

Rewrite the expression.

Phase Shift: $\frac{π}{4}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{3}$

Phase Shift: $\frac{π}{4}$ ( $\frac{π}{4}$ to the right)

Vertical Shift: $2$

Step 6

Find Amplitude, Period, and Phase Shift f(t)=4cot(3t-(3pi)/4)+2

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