Find Amplitude, Period, and Phase Shift y=4/3cot(4(x-pi/2))+1

$y = \frac{4}{3} \cot (4 (x - \frac{π}{2})) + 1$

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

$a = \frac{4}{3}$

$b = 4$

$c = 2 π$

$d = 1$

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

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

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Find the period of $\frac{4 \cot (4 x - 2 π)}{3}$ .

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

$\frac{π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{π}{4}$

Find the period of $1$ .

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

$\frac{π}{| b |}$

Replace $b$ with $4$ in the formula for period.

$\frac{π}{| 4 |}$

The absolute value is the distance between a number and zero. The distance between $0$ and $4$ is $4$ .

$\frac{π}{4}$

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

$\frac{π}{4}$

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: $\frac{2 π}{4}$

Cancel the common factor of $2$ and $4$ .

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Factor $2$ out of $2 π$ .

Phase Shift: $\frac{2 (π)}{4}$

Cancel the common factors.

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Factor $2$ out of $4$ .

Phase Shift: $\frac{2 π}{2 \cdot 2}$

Cancel the common factor.

Phase Shift: $\frac{2 π}{2 \cdot 2}$

Rewrite the expression.

Phase Shift: $\frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: None

Period: $\frac{π}{4}$

Phase Shift: $\frac{π}{2}$ ( $\frac{π}{2}$ to the right)

Vertical Shift: $1$

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