Find Amplitude, Period, and Phase Shift y=2cot(1/3x+pi/6)+2

$y = 2 \cot (\frac{1}{3} x + \frac{π}{6}) + 2$

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

$a = 2$

$b = \frac{1}{3}$

$c = - \frac{π}{6}$

$d = 2$

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

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

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Find the period of $2 \cot (\frac{x}{3} + \frac{π}{6})$ .

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

$\frac{π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 3$

Move $3$ to the left of $π$ .

$3 π$

Find the period of $2$ .

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

$\frac{π}{| b |}$

Replace $b$ with $\frac{1}{3}$ in the formula for period.

$\frac{π}{| \frac{1}{3} |}$

$\frac{1}{3}$ is approximately $0.\overline{3}$ which is positive so remove the absolute value

$\frac{π}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$π \cdot 3$

Move $3$ to the left of $π$ .

$3 π$

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

$3 π$

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: $\frac{- \frac{π}{6}}{\frac{1}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{6} \cdot 3$

Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{π}{6}$ into the numerator.

Phase Shift: $\frac{- π}{6} \cdot 3$

Factor $3$ out of $6$ .

Phase Shift: $\frac{- π}{3 (2)} \cdot 3$

Cancel the common factor.

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

Rewrite the expression.

Phase Shift: $\frac{- π}{2}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{2}$

List the properties of the trigonometric function.

Amplitude: None

Period: $3 π$

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

Vertical Shift: $2$

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Name

Name	nine hundred ten million two hundred sixty-one thousand eight hundred sixty-seven

Interesting facts

910261867 has 4 divisors, whose sum is 911368720
The reverse of 910261867 is 768162019
Previous prime number is 823

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 40
Digital Root 4

Name

Name	one hundred fourteen million four hundred nineteen thousand five hundred two

Interesting facts

114419502 has 16 divisors, whose sum is 305118720
The reverse of 114419502 is 205914411
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 27
Digital Root 9

Name

Name	six hundred seventy-two million five hundred eighty-five thousand eight hundred sixty-one

Interesting facts

672585861 has 8 divisors, whose sum is 902720704
The reverse of 672585861 is 168585276
Previous prime number is 151

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 48
Digital Root 3

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