Find Amplitude, Period, and Phase Shift tan(3x-pi/2)

$\tan (3 x - \frac{π}{2})$

Step 1

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

$a = 1$

$b = 3$

$c = \frac{π}{2}$

$d = 0$

Step 2

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

Amplitude: None

Step 3

Find the period of $\tan (3 x - \frac{π}{2})$ .

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 $3$ .

$\frac{π}{3}$

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: $\frac{\frac{π}{2}}{3}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{π}{2} \cdot \frac{1}{3}$

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

Phase Shift: $\frac{π}{2 \cdot 3}$

Multiply $2$ by $3$ .

Phase Shift: $\frac{π}{6}$

Step 5

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{3}$

Phase Shift: $\frac{π}{6}$ ( $\frac{π}{6}$ to the right)

Vertical Shift: None

Step 6