Find Amplitude, Period, and Phase Shift f(x)=tan(1/5x)

$f (x) = \tan (\frac{1}{5} x)$

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

$a = 1$

$b = \frac{1}{5}$

$c = 0$

$d = 0$

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 $\frac{π}{| b |}$ .

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

Period: $\frac{π}{| b |}$

Replace $b$ with $\frac{1}{5}$ in the formula for period.

Period: $\frac{π}{| \frac{1}{5} |}$

Solve the equation.

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$\frac{1}{5}$ is approximately $0.2$ which is positive so remove the absolute value

Period: $\frac{π}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Period: $π \cdot 5$

Move $5$ to the left of $π$ .

Period: $5 π$

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

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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: $\frac{0}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $0 \cdot 5$

Multiply $0$ by $5$ .

Phase Shift: $0$

Find the vertical shift $d$ .

Vertical Shift: $0$

List the properties of the trigonometric function.

Amplitude: None

Period: $5 π$

Phase Shift: $0$ ( $0$ to the right)

Vertical Shift: $0$

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