Find Amplitude, Period, and Phase Shift y=2sin((8pix)/3+(3pi)/2)

$y = 2 \sin (\frac{8 π x}{3} + \frac{3 π}{2})$

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

$a = 2$

$b = \frac{8 π}{3}$

$c = - \frac{3 π}{2}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $2$

Find the period of $2 \sin (\frac{8 π x}{3} + \frac{3 π}{2})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{8 π}{3}$ in the formula for period.

$\frac{2 π}{| \frac{8 π}{3} |}$

$\frac{8 π}{3}$ is approximately $8.3775804$ which is positive so remove the absolute value

$\frac{2 π}{\frac{8 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{8 π}$

Cancel the common factor of $2 π$ .

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Factor $2 π$ out of $8 π$ .

$2 π \frac{3}{2 π (4)}$

Cancel the common factor.

$2 π \frac{3}{2 π \cdot 4}$

Rewrite the expression.

$\frac{3}{4}$

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: $\frac{- \frac{3 π}{2}}{\frac{8 π}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{3 π}{2} \cdot \frac{3}{8 π}$

Cancel the common factor of $π$ .

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Move the leading negative in $- \frac{3 π}{2}$ into the numerator.

Phase Shift: $\frac{- 3 π}{2} \cdot \frac{3}{8 π}$

Factor $π$ out of $- 3 π$ .

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{3}{8 π}$

Factor $π$ out of $8 π$ .

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{3}{π \cdot 8}$

Cancel the common factor.

Phase Shift: $\frac{π \cdot - 3}{2} \cdot \frac{3}{π \cdot 8}$

Rewrite the expression.

Phase Shift: $\frac{- 3}{2} \cdot \frac{3}{8}$

Multiply $\frac{- 3}{2}$ and $\frac{3}{8}$ .

Phase Shift: $\frac{- 3 \cdot 3}{2 \cdot 8}$

Simplify the expression.

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Multiply $- 3$ by $3$ .

Phase Shift: $\frac{- 9}{2 \cdot 8}$

Multiply $2$ by $8$ .

Phase Shift: $\frac{- 9}{16}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{9}{16}$

List the properties of the trigonometric function.

Amplitude: $2$

Period: $\frac{3}{4}$

Phase Shift: $- \frac{9}{16}$ ( $\frac{9}{16}$ to the left)

Vertical Shift: None

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Name

Name	one billion seven hundred four million six hundred fifteen thousand six hundred eighteen

Interesting facts

1704615618 has 8 divisors, whose sum is 3409231248
The reverse of 1704615618 is 8165164071
Previous prime number is 3

Basic properties

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

Name

Name	one billion two hundred seventy million seven hundred fifty-seven thousand eight hundred ninety-seven

Interesting facts

1270757897 has 4 divisors, whose sum is 1286843520
The reverse of 1270757897 is 7987570721
Previous prime number is 79

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 53
Digital Root 8

Name

Name	one billion two hundred eighty-seven million two hundred sixty-two thousand five hundred thirty-five

Interesting facts

1287262535 has 16 divisors, whose sum is 1665002304
The reverse of 1287262535 is 5352627821
Previous prime number is 1223

Basic properties

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

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