Find Amplitude, Period, and Phase Shift y=4sin(2/3x+pi/9)

$y = 4 \sin (\frac{2}{3} x + \frac{π}{9})$

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

$a = 4$

$b = \frac{2}{3}$

$c = - \frac{π}{9}$

$d = 0$

Find the amplitude $| a |$ .

Amplitude: $4$

Find the period of $4 \sin (\frac{2 x}{3} + \frac{π}{9})$ .

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

$\frac{2 π}{| b |}$

Replace $b$ with $\frac{2}{3}$ in the formula for period.

$\frac{2 π}{| \frac{2}{3} |}$

$\frac{2}{3}$ is approximately $0.\overline{6}$ which is positive so remove the absolute value

$\frac{2 π}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$2 π \frac{3}{2}$

Cancel the common factor of $2$ .

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Factor $2$ out of $2 π$ .

$2 (π) \frac{3}{2}$

Cancel the common factor.

$2 π \frac{3}{2}$

Rewrite the expression.

$π \cdot 3$

Move $3$ to the left of $π$ .

$3 π$

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: $\frac{- \frac{π}{9}}{\frac{2}{3}}$

Multiply the numerator by the reciprocal of the denominator.

Phase Shift: $- \frac{π}{9} \cdot \frac{3}{2}$

Cancel the common factor of $3$ .

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Move the leading negative in $- \frac{π}{9}$ into the numerator.

Phase Shift: $\frac{- π}{9} \cdot \frac{3}{2}$

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Phase Shift: $\frac{- π}{3 \cdot 2}$

Simplify the expression.

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Multiply $3$ by $2$ .

Phase Shift: $\frac{- π}{6}$

Move the negative in front of the fraction.

Phase Shift: $- \frac{π}{6}$

List the properties of the trigonometric function.

Amplitude: $4$

Period: $3 π$

Phase Shift: $- \frac{π}{6}$ ( $\frac{π}{6}$ to the left)

Vertical Shift: None

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Name

Name	two hundred three million four hundred thirty-three thousand six hundred sixteen

Interesting facts

203433616 has 32 divisors, whose sum is 1029882762
The reverse of 203433616 is 616334302
Previous prime number is 2

Basic properties

Is Prime? no
Number parity even
Number length 9
Sum of Digits 28
Digital Root 1

Name

Name	one billion three hundred twenty-six million six hundred fifty-eight thousand four hundred thirty-five

Interesting facts

1326658435 has 8 divisors, whose sum is 1592302896
The reverse of 1326658435 is 5348566231
Previous prime number is 5717

Basic properties

Is Prime? no
Number parity odd
Number length 10
Sum of Digits 43
Digital Root 7

Name

Name	one billion nine hundred fifty-three million seven hundred fifty-four thousand three hundred ninety-eight

Interesting facts

1953754398 has 32 divisors, whose sum is 6946682496
The reverse of 1953754398 is 8934573591
Previous prime number is 3

Basic properties

Is Prime? no
Number parity even
Number length 10
Sum of Digits 54
Digital Root 9

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