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	three hundred ninety-five million five hundred fifty-eight thousand two hundred six

Interesting facts

395558206 has 4 divisors, whose sum is 593337312
The reverse of 395558206 is 602855593
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 43
Digital Root 7

Name

Name	one billion three hundred seventy-five million six hundred twenty-one thousand six hundred twenty-eight

Interesting facts

1375621628 has 16 divisors, whose sum is 3096868464
The reverse of 1375621628 is 8261265731
Previous prime number is 1817

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 41
Digital Root 5

Name

Name	six hundred forty-one million two hundred fifty-six thousand five hundred thirty-five

Interesting facts

641256535 has 8 divisors, whose sum is 769998096
The reverse of 641256535 is 535652146
Previous prime number is 1601

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 37
Digital Root 1

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