Find Amplitude, Period, and Phase Shift y=sin(6pix-(4pi)/3)

$y = \sin (6 π x - \frac{4 π}{3})$

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

$a = 1$

$b = 6 π$

$c = \frac{4 π}{3}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $1$

Find the period of $\sin (6 π x - \frac{4 π}{3})$ .

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The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 6 π |}$

$6 π$ is approximately $18.84955592$ which is positive so remove the absolute value

$\frac{2 π}{6 π}$

Cancel the common factor of $2$ and $6$ .

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Factor $2$ out of $2 π$ .

$\frac{2 (π)}{6 π}$

Cancel the common factors.

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Factor $2$ out of $6 π$ .

$\frac{2 (π)}{2 (3 π)}$

Cancel the common factor.

$\frac{2 π}{2 (3 π)}$

Rewrite the expression.

$\frac{π}{3 π}$

Cancel the common factor of $π$ .

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Cancel the common factor.

$\frac{π}{3 π}$

Rewrite the expression.

$\frac{1}{3}$

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: $\frac{\frac{4 π}{3}}{6 π}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $\frac{4 π}{3} \cdot \frac{1}{6 π}$

Cancel the common factor of $2 π$ .

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Factor $2 π$ out of $4 π$ .

Phase Shift: $\frac{2 π (2)}{3} \cdot \frac{1}{6 π}$

Factor $2 π$ out of $6 π$ .

Phase Shift: $\frac{2 π (2)}{3} \cdot \frac{1}{2 π (3)}$

Cancel the common factor.

Phase Shift: $\frac{2 π \cdot 2}{3} \cdot \frac{1}{2 π \cdot 3}$

Rewrite the expression.

Phase Shift: $\frac{2}{3} \cdot \frac{1}{3}$

Multiply $\frac{2}{3}$ and $\frac{1}{3}$ .

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

Multiply $3$ by $3$ .

Phase Shift: $\frac{2}{9}$

List the properties of the trigonometric function.

Amplitude: $1$

Period: $\frac{1}{3}$

Phase Shift: $\frac{2}{9}$ ( $\frac{2}{9}$ to the right)

Vertical Shift: None

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Name

Name	nine hundred forty-seven million seven hundred forty-three thousand four hundred twenty-seven

Interesting facts

947743427 has 4 divisors, whose sum is 997624680
The reverse of 947743427 is 724347749
Previous prime number is 19

Basic properties

Is Prime? no
Number parity odd
Number length 9
Sum of Digits 47
Digital Root 2

Name

Name	seven hundred ninety million ninety-nine thousand two hundred fifty-six

Interesting facts

790099256 has 16 divisors, whose sum is 2666585016
The reverse of 790099256 is 652990097
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 47
Digital Root 2

Name

Name	two billion sixty-two million eight hundred ninety-three thousand five hundred twenty-two

Interesting facts

2062893522 has 8 divisors, whose sum is 4125787056
The reverse of 2062893522 is 2253982602
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 39
Digital Root 3

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