Find Amplitude, Period, and Phase Shift f(x)=3+tan(6x+pi/6)

$f (x) = 3 + \tan (6 x + \frac{π}{6})$

Rewrite the expression as $\tan (6 x + \frac{π}{6}) + 3$ .

$\tan (6 x + \frac{π}{6}) + 3$

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

$a = 1$

$b = 6$

$c = - \frac{π}{6}$

$d = 3$

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

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

Find the period of $\tan (6 x + \frac{π}{6})$ .

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

$\frac{π}{| b |}$

Replace $b$ with $6$ in the formula for period.

$\frac{π}{| 6 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{π}{6}$

Find the period of $3$ .

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

$\frac{π}{| b |}$

Replace $b$ with $6$ in the formula for period.

$\frac{π}{| 6 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $6$ is $6$ .

$\frac{π}{6}$

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

$\frac{π}{6}$

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: $\frac{- \frac{π}{6}}{6}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot \frac{1}{6}$

Multiply $- \frac{π}{6} \cdot \frac{1}{6}$ .

Multiply $\frac{1}{6}$ by $\frac{π}{6}$ .

Phase Shift: $- \frac{π}{6 \cdot 6}$

Multiply $6$ by $6$ .

Phase Shift: $- \frac{π}{36}$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{6}$

Phase Shift: $- \frac{π}{36}$ ( $\frac{π}{36}$ to the left)

Vertical Shift: $3$